Originally published in: École d’Été de Probabilités de Saint-Flour XV-XVII –1985-87, Lecture Notes in Mathematics, Vol. 1362, 277–425, DOI: 10.1007/ BFb0086183, © Springer-Verlag Berlin Heidelberg 1988, Reprint by Springer-Verlag Berlin Heidelberg 2013
113
GEOMETRIC ASPECTS OF DIFFUSIONS ON MANIFOLDS
David ELWORTHY
114
D. ELWORTHY : "GEOMETRIC ASPECfS OF DIFFUSIONS ON MANIFOLDS"
Introduction
CHAPTER I: STOCHASTIC DIFFERENTIAL EQUATIONS AND MANIFOLDS
§ 1. Some notation, running hypotheses, and basic facts about manifolds. 282
§2. Stochastic differential equations on M . 285
§3. An Ito formula. 288
§4. Solution flows. 290
CHAPTER II : SOME DIFFERENTIAL GEOMETRY FOR PRINCIPAL
BUNDLES AND CONSTRUCTIONS OF BROWNIAN MOTION
§ 1. Connections on principal bundles and covariant differentiation. 296
§2. Horizontal lifts, covariant derivatives, geodesics, and a second form of the Ito formula. 299
§3. Riemannian metrics and the Laplace-Beltrami operator. 304
§4. Brownian motion on M and the stochastic development. 307
§5. Examples: Spheres and hyperbolic spaces. 312
§6. Left invariant SDS on Lie groups.
§7. The second fundamental form and gradient SDS for an embedded submanifold.
§8. Curvature and the derivative flow.
316
316
319
§9. Curvature and torsion forms. 323
§10. The derivative of the canonical flow. 327
CHAPTER III : CHARACTERISTIC EXPONENTS FOR STOCHASTIC FLOWS
§1. The Lyapunov spectrum. 331
§2. Mean exponents. 335
§3. Exponents for gradient Brownian flows: the difficulties of estimating exponents in general. 339
§4. Exponents for canonical flows. 342
§5. Moment exponents. 356
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CHAPTER IV : THE HEAT FLOW FOR DIFFERENTIAL FORMS AND THE
TOPOLOGY OF M
§ 1. A class of semigroups and their solutions.
§2. The top of the spectrum of ~.
§3. Bochner theorems for L2 harmonic forms.
§4. de Rham cohomology, Hodge theory and cohomology with compact support.
§5. Brownian motion and the components of M at infinity.
CHAPTER V: HEAT KERNELS: ELEMENTARY FORMULAE,
INEQUALITIES, AND SHORT TIME BEHAVIOUR
§ 1. The elementary formula for the heat kernel for functions.
§2. General remarks about the elementary formula method and its extensions.
§3. The fermionic calculus for differential forms, and the Weitzenbock formula
§4. An elementary formula for the heat kernel on forms.
CHAPTER VI : THE GAUSS-BONNET-CHERN THEOREM
§ 1. Supertraces and the heat flow for forms.
§2. Proof of the Gauss-Bonnet-Chem Theorem.
Bibliography
Notation
Index
360
363
367
371
374
379
391
398
402
407
408
413
422
423
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INTRODUCTION
A. There were three main aspects of the theory of diffusions on
manifolds presented in this course: the theory of characteristic
exponents for stochastic flows; the use of the Feynman-Kac formula for
the solution to the heat equation for forms to obtain geometric results,
in particular on the shape of the manifold at infinity; and a technique
for obtaining exact and asymptotic expansions of heat kernels. A proof
of the Gauss-Bonnet-Chern theorem was given as an application of the
third, and also to emphasize the fact that Malliavin calculus is not
needed for probabilistic proofs of the Atiyah-Singer index theorem.
The first two aspects concern long term behaviour and the third
short time behaviour.
Since the participants could not be expected to have a thorough
knowledge of differential geometry, quite a lot of time was devoted to a
quick course in Riemannian geometry via connections on the frame
bundle. Apart from the intuitive understanding this gives to the notion
of parallel translation along the paths of a diffusion, needed for the
Feynman-Kac formula for forms, it allows for a global formalism and is
anyway intrinsically involved in the main example considered in the
discussion of characteristic exponents. Also the fermionic calculus for
differential forms was described, both in order to give a proof of the
Weitzenbock formula: which is a basic result needed to be able to obtain
a Feynman-Kac formula, and in order to give the 'supersymmetric' proof
of the Gauss-Bonnet-Chern theorem. This was taken from [33].
Probabilistically it was assumed that the participants had a
reasonable understanding of stochastic differential equations in !R n ,
driven by Brownian motion, and with smooth coefficients. The existence
theorem for solutions of such equations on manifolds (up to an explosion
time) was proved by embedding the manifold in some !R n and then using
existence results for !R n . See also [1 07].
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B. Not surprisingly, perhaps, it was not possible to cover everything in
these lecture notes during the 15 hours of the course. The main sections
missed out were those relating to the geometry of submanifolds of IR n :
II §7, Ill §3; the section on moment exponents Ill §5; and much of V §2.
C. Acknowledgements. I would like to thank the participants for
their encouragement and helpful suggestions, as well as for their help in
pointing out the numerous errors in the first version of these notes. I
am particularly grateful to Monique Poitier for this. From my point of
view the whole summer school was extremely enjoyable as well as being
very stimulating, and I am very pleased to be able to express my
admiration of Professor Hennequin's efforts which made this possible. I
would also like to thank Steve Rosenberg for permission to use some
unpublished joint work in Chapter IV and to thank David Williams for
pointing out a mistake in my original version of Chapter VI §2C.
This final camera ready version was prepared by Peta McAllister
from the original version typed by Terri Moss: the difficulties they had
from a combination of my writing and changing technology were
considerable, and I owe them a big debt of thanks for their patience,
skill and speed.
Some of the work reported in the course was supported in part by
the U.K. Science and Engineering Research Council.
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CHAPTER 1
STOCHASTIC DIFFERENTIAL EOUA TIONS AND MANIFOLDS
§1. Some notatton, running hypotheses, and basic facts about manifolds
A. We will use invariant notation as much as posssible: if U is open in a
Banach space E and F is some other Banach space then the first derivative of a
function f: U -+ F will be written
Of : U -+ !I...(E,F)
and the second
o2f: u-+ !I...(E,E;F), etc.,
where !I...(E,F) and ll...(E,E;,F) are the spaces of continuous linear and bilinear
maps into F respectively. Sometimes we write
df : u X E-+ F
for
(x,v)-+ Df(x)(v)
with
(df)x for Df(x).
For continuous semi-martingales z1, z2 with values in finite dimensional
normed spaces E1, E2 and for bilinear Bt : E1 x E2 -+ F it will be more
convenient to use
T T
J J ij j j B1(dz1,t, dz2,t) for Li,j Bt d<zl'z2>t
0 0
or for some corresponding expression involving the tensor quadratic variation.
Here the i,j refer to co-ordinates with respect to bases of E1, E2 respectively.
Thus
o2f(x)(dz1,t• dz2,t) = Li,j a2f /oxiaxi d<zi 1• zi2>t
for E = Ei = IRn.
B. Our manifolds M will be coo, connected, and of finite dimension n, unless
clearly otherwise. They will always be metrizable. The manifold structure is
determined by some coo atlas ((Uex, fPex) : ex E A where A is some index set,
(Ucx : ex E A is an open cover, and each fPcx is a homeomorphism of Uex onto
some open subset of IRn, such that on their domain of definition in IRn each
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coordinate change <9o: o <9.13-1 is coo. The pairs (Uo:, <9o:) are C00 charts, as
are any other such which when added to the original atlas still keeps it a coo
atlas. For 0 :$ r :$ oo maps f: M-+ N of manifolds are cr if each ep o f o <9o:-1
is cr on its domain of definition when (Uo:, <9o:) is a chart forM and (Vp, ep)
one for N. The map is a cr diffeomorphism if it is a homeomorphism and both
it and its inverse are cr. The spaces lRn are considered as coo manifolds by
taking ((lRn, identity map)) as atlas, and similarly for open sets U of lRn. A
subset N of [Rn+p is an n-dimensional coo submanifold if there is a family of
coo charts ((UJ3, cp.13): J3 e B) for lRm such that UJ3 covers M and IPJ3-1 (lRn x (O)
= N n U.13 for each .13 e B (writing [Rn+p = lRn x lRP). Then ((U.13 n N, IPJ31UJ3 n N):
J3 e B) forms an atlas for N, making it a coo manifold.
A coo map f : M -+ [Rn+p is an embedding if its image N := f(M) is a coo
submanifold of [Rn+p and f gives a diffeomorphism of M onto N.
C. A tangent vector at x e M can be considered as an equivalence class of
smooth curves
~: (-£,£)-+ M
some £ > 0 with ~(0) = x where ~1- ~2 if d/dt <9o:(~1(t)) and
d/dt ~Po:(~2(t)) agree at t = 0 for some (and hence all) charts (Uo:, ~Po:) with
x e Uo:. The set of all such forms the tangent space T xM to M at x. Any chart
around x gives a bijection
T x~Po: : T xM -+ lRn
which is used to give T xM a vector space structure (independent of the choice
of chart). When M is lRn itself, or an open subset of lRn, the tangent space is
naturally identified with lRn itself. A cr map f : M-+ N of manifolds r ~ 1
determines a linear map
T xf : T xM-+ T f(x)N
(obtained by considering the curves f o ~for example).
The disjoint union TM of all tangent spaces (T xM: x e M) has a projection
"t: TM-+ M defined by "t-1(x) = T xM. This is the tangent bundle: it is a coo 2n
dimensional manifold with atlas
("t-1(Uo:), T<po:) : o: E A)
where
T<po:(v) = (cpo:(x), T x <9o:(v)) e lRn x lRn
for v e T xM, for ((Uo:, <9o:) : o: e A an atlas of M. Our cr map f : M-+ N
determines a cr-1 map Tf : TM-+ TN where Tf restricts to T xf on T xM. The
assignment of Tf to f is functorial. Note that for U open in lRn we can identify
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TU with U x IRn.
For f: M-+ IR write df: TM-+ IR for v-+ T "t(v)f.
D. When f: M-+ IRn+p is an embedding each T xf: T xM-+ IRn+p is injective, and
Tf: TM-+ IRn+p x IRn+p, namely v-+ (x,T xf(v) for veT xM, is an embedding. This
is used to identify T xM with its image in IRn+p.
Now suppose M is a closed submanifold of IRn+p (i.e. a closed subset,
not necessarily compact): the identity map is an embedding and we make the
above identifications. There is the normal bundle v(M) toM:
v(M) = (x,v) eM x IRn+p: v ~ T xM.
Then v(M) has the obvious projection onto M with fibres vx(M) : x e M linear
spaces of dimension p. By considering the map
P : M x IRn+p -+ T xM
P(x,v) = Px(V) E T xM
where Px is the orthogonal projection of IRn+p onto T xM, use of the implicit
function theorem shows that v(M) is a submanifold of IRn+p x IRn+p.
Next we will construct a tubular neighbourhood of M in IRn+p; this will be
used to save worrying about details in the proof of the existence of solutions
of S.D.E. The exponential map of IRn+p is the map
exp : TIRn+p = IRn+p x IRn+p -+ IRn+p
exp(x,v) = x + v.
Restrict it to v(M) to get lj) : v(M)-+ IRn+p• say. Differentiate at (x,O) to get
T(x,O)IP: T(x,O) v(M)-+ IRn+p_ Now T(x,O) v(M) consists of the direct sum of
T xM (the "horizontal" part) and the tangent space to vx(M) at 0 which can be
identified with Vx(M) itself since it is a linear space. With this identification
T(x,o)IP(v,w) = v + w
so that T(x,O)IP is a linear isomorphism. The inverse function theorem
(applied in charts) implies then that lj) restricts to a coo diffeomorphism of a
neighbourhood of (x,O) in v(M) onto an open neighbourhood of x in IRn+p.
Piecing these together for different x in M we see there is an open
neighbourhood of the zero section (x,O) e v(M) : x e M, which is mapped
diffeomorphically onto an open neighbourhood of M in IRn+P. In particular
(using a coo partition of unity on M) there is a smooth function a : M-+ (0, oo)
such that t.p gives a diffeomorphism lj) :(x,v) e v(M) :I v I< a(x)-+ Na(M) where
Na(M) = UxeM y e IRn+p s.t. I y-x I< a(x). This will be called the tubular
neighbourhood of M radius a. Note that on Na(M) the map giving the distance
squared from M
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y-+ d(y,M)2
is C00 since d(y,M) =I projection on the second factor of ~.p-1(y) 1. Examples
to think of are the spheres sn in 1Rn+1 and the surface of revolution z = (x2 +
y2)-1, for x,y positive, in IR3.
§2. Stochastic differentfal equations on M
A. For a probability space (Q,~,IP) consider a filtration ~t: t ~ 0, assuming
for simplicity that each ~ t contains all sets of measure zero in ~.
Let Y: M x IRm-+ TM be c2 with each Y(x)(-) := Y(x,-) linear from IRm into
T xM· For t 0 ~ 0 let zt, t ~ t 0 be a continuous semi-martingale on IRm and let
u:Q-+M
be ~t -measurable. By a solution to the stochastic differential equation 0
dXt = Y(Xt) o dZt t ~ t 0 (1)
with x1 = u 0
we mean a sample continuous, adapted process,
Xt : Q -+ M t0 s t < s where s : Q-+ [O,oo] is a stopping time such that for any c3 map f : M-+ IR and
stopping timeT with t 0 :S T < s there is the Ito formula
T
f(xT) = f(u) + J T x f(Y(xs) o dzs) \) s
a.s.
where o denotes Stratonovich integral.
(2)
B. The pair (Y,z) will be called a stochastic dynamical system (S.O.S.). Write
(to.s) X Q for (t,w) E [to,oo) X Q: t < s(w). When M = IRn this is equivalent to
the usual definition and there is the basic existence and uniqueness theorem:
Existence and Uniqueness Theorem for IR"
Given z,Y, and u as above forM= IRn there is a unique maximal solution
to (1)
defined up to an explosion time ~ which is almost surely positive. On
(~ < oo, as t t ~(w) so Xt(w)-+ oo in IRn a.s.
Uniqueness holds in the sense that if (Yt: t0 :S t < S is any other solution
starting from u at time t 0 then s s ~ almost surely and x I [t 0 ,s) x Q = y
almost surely. Indeed this is easily deduced from the usual case of an Ito
equation with globally lipschitz coefficients: for R = 1,2, ••• take a C00 map <pR :
122286
IRn-+ IR(> 0) identically one on the ball B(O;R) about 0 radius R and with
compact support. Set YR =<pRY . In its Ito form
dXt = YR(Xt) o dZt
has Lipschitz coefficients and has a solution xR, say, defined for all t ~ t0 ,
starting at u at time t 0 • If SR is its first exit time from B(O;R) then XR' agrees
with xR up to time SR if R' > R , and so a limiting solution x to (1) can be
constructed as required.
C. Existence and uniqueness theorem for M: We will now see why the same
holds for a general manifold M. The straightforward proofs of this tend to be
either tedious or unsatisfying, so we will try to avoid tedium by some
geometrical constructions.
First of all for there to be uniqueness there clearly has to be an ample
supply of c3 functions f. Since we will have to use that fact we may as well
use a strong form of it, namely: Whitney's embedding Theorem. There is a
C00 embedding <p : M-+ IRn+p of M onto a closed submanifold of IRn+p where
p=n+1.
The exact codimension p will not be important, and the proof then, for
some p, is a straighforward argument using partitions of unity and the implicit
function theorem.
Taking such an embedding <p, identify M with its image, so that we can
consider it as a submanifold of IRn+p. It is easy, using C00 partitions of unity
to extend Y to a c2 map Y: IRn+p x IRm-+ IRn+p x IRn+p. However we will do a
more explicit extension below. Since a c2 function on IRn+p restricts to a c2
functlon on M a solution to (1) is a solution to
(I)
in IRn+p (strictly speaking, if Xt satisfies (I) then <p(Xt) satisfies (1) since
f =I o <pis c3). Thus uniqueness holds for (1) since it does for (I). Conversely
any c3 map f : M-+ IR can be extended to a c3 map I: IRn+p-+ IR , so a solution
to (I) which lies on M for all time must solve (1). To prove existence of a
maximal solution it is enough therefore to show that if xt : t 0 :S t < S is the
maximal solution to (1) with Xt = u for u : Q-+ M then Xt lies in M almost 0
surely for all t. In fact it is enough to do that for just one extension Y. In
particular take a tubular neighbourhood of M radius a: M-+ IR(> O) as described
in §1D; chooseR> 0 and let
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aR = inf a(x): X eM n B(O;R + 1)) > 0.
Choose smooth)..: IRn+p-+ 1R(2: O) with support in B(O;IR+1) and identically one
in B(O,R), and smooth Jl : [O,oo)-+ 1R(2: 0) with J.l(X) = 1 for I x Is ~ aR2 and,
J.l(X) = 0 for I x I> aR2 •
Let TI : Na(M)-+ M map a point to the nearest point of M to it. Using the
identificatlon tp of Na(M) with part of v(M) this is seen to be C00 • Define
YR: IRn+P x IRm-+ IRn+p by
YR(x)(e) = 0 for x ~ Na(M), e e IRm
YR(X)(e) = J...(X)jl(d(x,M)2) Y(TI(x))(e) for X E Na(M), e E IRm.
Define f: IRn+p-+ IR 2: ·0 by f(x) = J...(X)J.l(d(x,M)2).
Inside B(O;R) the map f is constant on the level sets of d(-,M) while
YR(x)(e) is tangent to these sets. Therefore
Df(x) (YR(x)e) = 0 X E B(O,R), e E IRm
and so any solution Xt: t 0 s t < S) to (r) starting on M satisfies f(Xt) = 0
almost surely until it first exits from B(O;R). It therefore stays on M almost
surely until this time. Since R was arbitrary we are done:
The uniqueness and existence up to an explosion time result of §28 holds
exactly as stated when IRn is replaced by a manifold M.
D. It is now easy to see:
If N is a closed submanifold of M and Y(x)e lies in T xN for all points x
of N then any solution to (1) which starts on N almost surely stays on N
for its lifetime.
Indeed we now know there is a solution to the restriction of (1) toN which
exists until it goes out to infinity on N, or for all time. This solves (1) on M
and is certainly maximal: it must therefore be the unique maximal solution to
1).
E. Usually our equations will be of the form
dXt = X(Xt) o dBt + A(Xt)dt (1)'
where X : M x IRm-+ TM is as Y was before, (Bt: t 2: O) is a Brownian motion on
IRm, and A is a vector field on M, so A(x) e T xM for each x eM. This fits into
the scheme of (1) with Zt = (Bt,t) e IRm x IR = 1Rm+1 and Y(x)(e 1,r) = X(x)(e 1) +
rA(x) when (e 1,r) e IRm x IR. However in this case it is of course only
necessary to assume that X is c2 and A is c1.
m
For an orthonormal base e1, ••. ,em of IRm write Bt as L gr t ero so the r=l
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Brt: t 2: 0, r = 1,2, ... ,m are independent Brownian motions on IR. Let x1, ... ,xm
be the vector fields
XP(x) = X(x)(ep)
Then (2) becomes
§3. An Ito formula
p = 1, ... ,m.
(2)
A. We will need (2) in Ito form. One version using covariant derivatives will
be given below in § 2 of Chapter II. However it will be useful to have a form
which does not depend on a choice of connection for M e.g. when we need to
consider equations on pri nci pal bundles it would be a nuisance to have to
describe some way of covariant differentiation of vector fields on principal
bundles.
For equation (1) and each e e lRm let
( t,x) -+ S( t,x)e
be the flow of the vector field Ye := Y(-)(e) on M, defined on some
neighbourhood of 0 x Min IR x M.
A vector field, e.g. Y e• acts on f : M -+ IR to give
Yef:M-+IR
by Y e f(x) = df(Y e(X) ).
Thus, for our c2 function f
and
d/dt f(S(t,x)e) = df(Ye(S(t,x)e))
= Yef (S(t,x)e)
d2/dt2 f(S(t,x)e) = Ye Yef(S(t,x)e).
At t = 0 this gives linear and bilinear maps which we write
d/dt foS(t,x) I E IL(IRm;IR) t=O
and
d2/dt2 foS(t,x) I E IL(IRm, IRm;IR) t=O
(3)
(4)
Propos1t1on 3A (Global Ito formula) Far a solution xt: 0 :S t < S of (1), if
f: M-+ IR is c2 and Tis a stopping time less than~ then, using Ita integrals,
almost surely:
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T T
f(xr) = f(x 0 ) +J d/dt f•S(t,xr)l (dzr) + ~J d2 /dt2 f•S(t,xr)l (dzr,dzr) (5) 0 t=O 0 t=O
In another form: if e 1, .•• ,em is a base for IRm and
then
T T
f(xr) = f(x 0 ) + J LvP f(xr)dzPr + ~ J L vP Yqf(xr) d(zP,zq>r (6) 0 p 0 p
Proof. If f is c4 the Stratonovich correction term for (2) is
T
~ J L dvPt dzqt where vPt = df(YP(x1)) = YPf(x1), 0 p,q
and (6), and hence (5), follows by calculating dvPt using (2) applied to YP(t).
For f only c2 the easiest way to proceed is to embed M in some IRn+p, extend f
to some c2 function f, extend Y toY as before and write
in Ito form. Then apply the usual Ito formula to f to obtain (6) after
restriction. //
B. For equation (2)'
dXt = X(Xt) o dBt + A(Xt)dt
form (6) becomes
T T
f(XT) = f(x 0 ) + J df(X(Xr)) o dBr + J .A. f(Xr)dr 0 0
where
m
.A.f = ~ L xPxPr + Af. p=l
(6)'
Using the results for IRn+p after embedding M in IRn+p we see the
solutions of (2)' form a Markov process with differential generator .A. •
C. A sample continuous stochastic process Yt : 0 ::s t < S) on M is a semi
martingale if f(y1) : 0 ::s t < 0 is a semi-martingale in the usual sense
whenever f : M..,. IRis c2, (Schwartz [90]). The above formulae show that our
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solutions x1
to (1) are semi-martingales. There is also the converse result,
observed by Schwarz in [90], every continuous semi-martingale y on M is the
solution of some equation like (1): indeed given y take some embedding cp: M-+
IRn+p, some p. Let Zt = cp(yt)· Then z is a semi-martingale, and if P: M x IRn+p
-+ TM is the orthogonal projection map, as in §10, then y is a solution to
dxt = P(xt) o dzt (7)
One easy way to see this is to use the projection TT : Na(M)-+ M of a tubular
neighbourhood as in §1D: then, for Xt = Yt• equation (7) is the differential form
of the equation n(zt) = Zt·
D. For a cl map f: M-+ N of manifolds, stochastic dynamical systems (X,z) on
M and (Y,z) on N are said to be f-related if T xf(X(x)e) = Y(f(x))e for x eM and
e e IRm. The corresponding result for O.D.E. together with equation (5)
immediately shows that it f ts c2 and Cxt : o :::::; t < 0 is a solution to dxt =
X(Xt) o dZt then f(Xt): 0:::::; t < S is a solution to d!:Jt = Y(yt) o dZt on N. For a c3
map f it is immediate from the definition.
§4. Solut1on flows
A. Flows of stochastic dynamical systems were discussed in Kunita's
Stochastic Flow course in 1982 , [67], and I do not want to go into a detailed
discussion of their existence and properties. However I would like to describe
briefly a method which gives the main properties of these· flows rather
quickly, and also mention some rather annoying gaps in our knowledge. The
first problem is to find nice versions of the map
(t,x,w)-+ Ft(x,w) eM we Q
which assigns to x e M the solution to the S.D.E. starting at x at time 0. In fact
throughout this section we assume Zt = (Bt,t) e IRm+l so that we are really
dealing with (1)'.
B. ForM compact, or more generally for Y of compact support, or for
M = IRn with Y having all derivatives bounded it is not difficult to show that
Totoki's extension of Kolmogorov's theorem can be used to obtain a version ofF
such that
(i) For all X EM, Ft(X,-): t ~ 0) solves dXt = Y(Xt) o dZt with F0 (x,-) =X.
(ii) Each map [O,oo) x M-+ M given by
(t,x) -+ F t(x,w) is continuous.
This was the method used by Blagovescenskii and Freidlin, for example see
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[43], [59], [9], [67], [78]. The extension of Kolmogorov's theorem used is:
Let (M,d) be a complete metric space. Suppose for x : [0,1] x ••• x [0,1]
(p-times) -+ t 0 (Q, ~;M) there exists ex, j3, 6 > 0 such that for all o > 0 and
s,t in [0,1]P
IP d(xs,Xt) > & s j3o -ex Is-tiP+'¥
then x has a sample continuous version.
The necessary estimates are most easily obtained by embedding Min some
IRn+p and extending Y as before.
C. For M compact it is possible to obtain differentiability, diffeomorphism,
and composition results by considering an induced stochastic differential
equation on the Hilbert manifold of Hs diffeomorphisms of M for s > ~n + 3: the
solution of this equation starting at the identity map being a version of Ft, see
[43]. Rather than discuss the Hilbert manifold structure of these groups of
diffeomorphisms it is possible to embed M in some IRn+p and extend the S.D.S.
over IRn+p as before, to have compact support. A now for the extended system
will restrict to one for the system of M. For the extended system we consider
the space of diffeomorphisms of class Hs of IRn+p which are the identity
outside of a fixed bounded domain U, containing the support of the extended
system. We will describe this rather briefly, see [240] for details.
Suppose therefore there is the system on IRn
dXt = Y(Xt) o dZt
where Y has compact support in U, for U open, bounded, and with smooth
boundary.
For s > n/2 set
Hsu(IR 0 ; IRP) = f e Hs(IRn;IRP) with suppf c U ) where Hs(IRn;IRP) is the
completion of the space C000 (1Rn; IRP) of C00 functions with compact support
under < >s where
(f,g)s = ~ JIRn (Dexf(x), oexg(x))dx
where the sum is over multi-indices ex = (ex 1 , ... ,exn) with oex =
I I ex1 exn a ex /(ox 1 .•• oxn ). Because s 2: n/2 the evaluation map x -+ f(x) is
continuous on Hs and so Hsu is a well defined closed subspace of Hs.
For s > n/2 + 1 set
.t'lsu = f : IRn-+ IRn s.t. f is a c1 diffeomorphism and 1d-f e Hsu(IRn;IRn) where
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1d refers to the identity map. Since diffeomorphisms are open in the c1
topology llsu is an open subset of the affine subspace 1d + Hsu(IRn;IRn). It is
therefore a Hilbert manifold with chart f-+ f - 1d. Moreover
1. llsu is a topological group under composition
2. For hE llsu right multiplication
Rh : llsu-+ llsu
f-+foh
is C00 •
3. Fork= 0,1,2, ••• composition
<pk : lls+ku x llsu -+ llsu
(f ,h) -+ f 0 h
is ck.
4. For k = 1,2, .•. inversion h-+ h -1 considered as a map
:Jk :lls+ku-+ llsu
is ck.
Now define the right invariant stochastic dynamical system
dht = Y(ht) o dzt
on llsu, always assuming s > n/2 + 1, by taking
Y(h) : IRm -+ T h llsu ==< Hsu h e llsu
to be
Y(h)(e) = <p2(Y(-)(e),h)
treating x-+ Y(x)(e) as in Hus+2. Thus Y is c2 by (3) above. We see
Y(1d)(e) = Y(-)(e)
and
Y(h)(e) = DRh(ld)(Y(Id)e)
so that Y is right invariant.
Proposltjon 4C. The solution ht to dht = Y(ht) o dzt starting from 1d
exists for all time, and Ft(-,w) = ht(w)(-) gives a flow for our equation on IRn
lying in J:lSu· In particular there is a C00 version of this flow, such
that t-+ F1(-,w) is almost surely continuous into the C00 topology.
Proof: A maximal solution certainly exists up to some predictable stopping
tlme s say, with s > 0 almost surely: the theory for equations of this type on
open subsets of a HHbert space goes through just as in finite dimensions,
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(provided one always uses uniform estimates, i.e. uses basis free notation).
Choose a predictable stopping t1 me T with 0 :S T :S s. By the right 1 nvari a nee of
the system, for h' e llsu
(s,w)-+ hs(w).h'
is a solution starting at h', so
(s,w)-+ hs-T(w)<eT(w))hT(w)(w)
where eT is the shift, is equivalent to hs(w) for s > T(w). Thus
s(w) 2::: T(w) + T(eT(w)).
Iterating this
.. s(W) 2::: k T (ekT(w)).
co Since Toek-rlk;O is an i.i.d. sequence we see s = oo almost surely. To see that
ht(w)(x0 ) is a solution to the equation on IRm starting at x0 observe that the two
equations are f-related for f the evaluation map f(h) = h(x0 ), and use the result
of §30. Finally since s > n/2 + 1 was otherwise arbitrary we see that almost
surely s-+ hs(w) lies in the space of continuous maps into
a J)SU With 1ts induced topology: but this is just a subset Of the space Of
C00 diffeomorphisms, with its relative topology. II
By embedding we deduce the existence of a C00 flow of diffeomorphisms
F1(-,w): M-+ M continuous in t into the C00 topology when M is compact.
Results like those of Bismut and of Kunita on the stochastic differential
equations for the inverses of the flows and compositions are easily obtained
via the inflnite dimensional Ito formula, see [24].
D. For non-compact manifolds M we can embed M and approximate the extended
S.O.S. by equations with compactly supported coefficients as in §1F. This
gives [66], [24a] Kunita's results on the existence of partial flows:
There is an explosion time map~ : M x Q-+ (O,oo] and a partially defined
flow F for dx1 = Y(xt) • dzt such that if
M(t)(w) = x EM: t < ~ (x,w) we Q
then for all w e Q
(i) M(t)(w) is open in M i.e. S(-,w) is l.s.c.
(i1) Ft(w) : M(t)(w)-+ M is defined and is a C00 diffeomorphism onto an open
subset of IRn
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(iii) For each x in M, )(x) is a stopping time and (Ft(X,-) : o :s; t < )(x)) is
a maximal solution. Moreover if K is compact in M, if S(K)(w) = inf
()(x)(w):x e K) then on )(K)(w) < oo, for any x0 eM, sup d(x0,Ft(x,w))-+ xEK
oo almost surely as t t )(K)(w).
(iv) the maps-+ Fs(-)(w) is continuous from [O,t] into the C00 topology of
functions on M(t)(w).
When we can choose s = oo the system is said to be strongly complete, or
strictly conservative. The standard example of a system which is complete
but not strongly complete is
dXt = dBt
on M = !R2 - (0). It is complete since Brownian motions starting outside of 0
in IR2 almost surely never hit 0. However the flow would have to be
Ft(X,W) =X+ Bt(W)
so that
s(x,w) = inf (t : Bt(W) = -X).
Using the fact that the solution to a complete system exists for all time
even when starting with a given random variable x0 : Q-+ M (independent of the
driving motion of course) it follows [27] that given completeness
J1 ® IP((x,w) : s(x,w) < oo) = 0
for all Borel measures J1 on M. In particular for almost all w £ Q each open set
M(t)(w) has full measure in M.
Even when the system is strongly complete the maps Ft(-,w) may not be
surjective: the standard example is the 2-dimensional Bessel process on M = (O,oo), see [43], [44], [59]. It seems difficult to decide when strong
completeness holds. It depends on Y, not just on the generator A: as observed
by Carverhill the Ito equation
dZt = (Zt/1 Zt I) dBt
on a::: - (O) is strongly complete (using complex multiplication for 8 a
Brownian motion on a:::::::: IR2), yet its solutions are just Brownian motions on
IR2 - (0). A general problem is therefore to find conditions on A and M
which ensure that there exists a choice of Y , giving a diffusion with
generator A , which is strongly complete. Anticipating some concepts from
the next chapter: for a complete Riemannian manifold M does a lower bound
on the Ricci curvature ensure strong completeness of the canonical SDS on
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OM? What conditions on M ensure that it admits a strongly complete SDS with
A = ~~ ? More generally is strong completeness of Y implied by the existence
of a uniform cover for Y in the sense of [43], [24a]?
E. Assume now that the system is complete. Even if it is not strongly
complete we can formally differentiate solutions in the space directions, to
get "derivatives in probability" rather than almost sure derivatives, [43]. This
was done forM= IRn in [57] to get L2-derivatives given strong conditions on Y.
These agree with the almost sure derivatives in the derivatives of the flow
F1(-,w) where the latter exists, and we will write them here as T x F1(-,w): 0
T x0
M--. T Xt(w)M, as if the flow did exist.
For veT x M set Vt = T x Ft(v,-). Then, either by embedding in some !Rn+p 0 0
or by arguing directly, (vt: t 2: 0) satisfies an equation
dv1 = &Y(v1) o dzt
on TM, where now &Y: TM x IRm--. T(TM) and is given by
&Y(v)e = ex TYe(v)
for Ye = Y(-)e : M -+ TM and ex : TTM -+ TTM the involution which over a chart
which represents TU as U x IRn and so TTU as (U x IRn) x (IRn x IRnl is given by
(x,u,v,w)-+ (x,v,u,w).
Thus in these co-ordinates
o Y( (x,u))e = (x,u, Y(x)e,DY e(x)( u)).
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CHAPTER II
SOME DIFFERENTIAL GEOMETRY FOR PRINCIPAL BUNDLES AND CONSTRUCTIONS
OF BROWNIAN MOTION
§1. Connections on principal bundles and covariant differentiation
A. A Lie group G is a C00 manifold with a group structure such that the maps
GxG-+G and
are C00 • Standard examples include the circle s1, the three sphere s3 (the
multiplicative group of quaternions with unit norm), orthogonal groups O(n)
which have SO(n) as connected component of the identity, and non-compact
groups (IRn, +)and GL(n).
A right action of G on a manifold M is a C00 map M x G-+ M
usually written (x,g)-+ x.g such that x.1 = x (for 1 the identity element), and
(x.g1)·92 = x.(g1.g 2 ). Examples are the action of G on itself by right
multiplication. The natural action of GL(n) on IRn is a left action, defined
similarly. Note that x-+ x.g is a diffeomorphism of M, which we will write as
Rg: M-+ M, with Lg : M-+ M for a left action.
Let g be the tangent space to G at 1. Then the left action gives a
diffeomorphism (trivialization of TG)
y: G X g-+ TG
Y(g)(v) = T 1 Lg(v).
Thus a semi-martingale z on g gives an S.D.E. on G.
The map v-+ Yv := Y(-)(v) gives a bijection between g and the space of
left-invariant vector fields on G. For any two vector fields x1, x2 on a
manifold M there is another vector field [x1, x2], the Lie bracket, determined
by [xl, x2]f = x1x2r- x2x1r for f: M-+ IR a c2 function. This gives a Lie
algebra structure. For a Lie group the bracket of two left invariant vectors
remains left invariant, so there is an induced Lie bracket on g s.t.
Y[v1,v2] = [Yv1,Yv2].
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Let <5(t): t e IR) be the curve in G with <5(0) = 1 and d<5/dt = Yv(<'(t)).
It is a 1-parameter subgroup and usually written o(t) = Exp(tv). This
determines
Exp: g~ G
b!J v ~ Exp v, not to be confused with other exponential maps (e.g. in § 1D and
§28 below).
B. A principal G-bundle over M is a map TT: 8 ~ M of differentiable manifolds
which is surjective, where B has a right G-action s.t. n(b.g) = n(b) all be B,
g e G, and such that there is an open cover (Ucx: : ex: e A of M for which there
exist C00 diffeomorphisms (local trivializations)
ecx:: n-1cucx:) ~ Ucx: X G
of the form
with
ecx:,x(b.g) = ecx:,xCb ).g.
X = TT(b)
The simplest example is the product bundle M x G with obvious right
action. An important example is the full linear frame bundle of M , TT : GLM
~ M, where GLM consists of all linear isomorphisms u : IRn ~ T xM for some x e
M, with TT mapping u to the relevant x. The right action is just composition
u.g(e) = u(ge) for e e IRn. Each element u is called a frame since it can be
identified with the base (u1•···•ur1) of T xM where Up = u(ep) for e1, ... ,en the
standard base for IRn.
Given principal bundles n i : Bi ~ M with groups Gi for i = 1,2 a C00
homomorphism is a C00 map h: s 1 ~ s2 such that h(b 1.g 1) = h(b 1).h 0 (g 1) for
some smooth group homomorphism h0 : G1 ~ G2. Thus every principal bundle
is by definition locally isomorphic to the trivial(:= product) bundle, (with h0 =
identity map).
C. For a principal G-bundle n : 8 ~ M the tangent space TB has a naturally
defined subset: the vertical tangent bundle, or bundle along the fibres,
VTB = v e TB: Tn(v) = 0.
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A connection on 8 is an assignment of a complementary "horizontal" tangent
bundle, HT8, invariant under the action of G. One way to do this is to take a g
valued 1-form ro i.e.
ro : T8-+ g
is smooth, and each restriction rob: Tb8-+ g, bE 8 is linear, with
(i) roo TRg = ad(g-1) oro g E G (8)
where ad(g-1): g-+ g is the adjoint action, namely the derivative at 1 of the
map G-+ G, a-+ g-1ag, and
(ii) ro(A*(b)) =A bE 8, A E g (9)
where A* is the (vertical) vector field on 8 defined by
A *(b) = d/dt (b. exp tA) lt=o·
Such an ro is called a connection form. Given such,one can define a
horizontal tangent bundle by
HT8 = v E T8: ro(v) = 0).
Then
(a) Tb8 = HTb8 EB VTb8
and
(b) TRg(HTb8) = HTb.g8
each bE 8
bE 8, g E G.
Conversely given HT8 satisfying (a) and (b) and smooth there exists a
connection form inducing it. It is easy to construct connections by partitions
of unity: but without additional structure there is no canonical choice.
For each trivialization (Uo:, eo:) there is a local section
so:: uo:-+ n-1cuo:) c 8
so:(x) = e-1o:,xC1)
This can be used to pull back a connection form ro to a g-valued 1-form roo: =
so:*(ro) given by the composition roo:= rooTso:: TUo:-+ g. For a connection on
GL(M) the components of this will give the Christoffel symbols: indeed a chart
(Uo:,fPo:) for M determines a trivialization which maps u to (T x fPo:) o u: !Rn-+
!Rn for a frame u at x. Then so:(x) = (T xfPo:)-1: !Rn-+ T xM. Define
f' : fPo:(Uo:)-+ IL(IRn;g) = IL(IRn;L(IRn;IRn))
by
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f'C<J'cx(x))v = scx*(ro)(T x\Pcx - 1(v)) v E IHn
giving the classical Christoffel symbols rijk
ri jk(Y) = (r(y)( ej)(ek),ei >
(10)
( 11)
where e1, ... ,en is the standard base for IRn. The connection is torsion free if
r(y)(v1)(v2) = r(y) (v2)(v1) or equivalently if rijk = r\j·
§2. Horizontal lifts, Covariant derfvatives, geodesics and a second form of the
Ito formula.
A. A connection for n: 8-+ M determines a horizontal lifting map
Hb : T n(b)M-+ T b8
which is the inverse of the restriction of Tbn to HTbB, a linear isomorphism.
It also gives a way of horizontally lifting smooth curves in M to curves in 8; a
smooth· curve <5 in 8 is horizontal if cr(t) E HT8 for all t or equivalently if
w(cr(t)) = o all t:
For a piecewise c1 curve <5: [O,T)-+ M and b0 E n-1(<5(0)) there exists a
unique horizontal curve <5- in 8 with <5-(o) = b0 and n(<5-(t)) = <5(t) for all
t (i.e. <5- is a lift of <5).
In fact given a trivialization (Ucx,ecx), while <5(t) is in Ucx, for <5- to be a
lift ecx(<5-(t)) has to have the form
ecx(<5-(t)) = (<5(t),g(t)) e Ucx x G
and then it will be horizontal if and only if
d/ dt g( t) = - TRg(t) (fi\x< <5( t))(cr( t))) ( 12)
so that local existence and uniqueness follow immediately from standard O.D.E.
theory on G. To see how (12) arises note that the axioms for 6i imply that for
(v,A) in T(x,a) (Ucx x G)
ffio(T9cx)-1(v,A) = ad(a-1)fficx(x)v + TL -1A (13) a
Note that by uniqueness and invariance, if g E G the horizontal lift of <5
starting at b0 .g is just t-+ <5-(t)g.
The only real difficulty in extending this construction to lifts of semi
martingales Yt (e.g. by treating (12) as a Stratonovich S.D.E. when <5(t) is
replaced by y1) is to make sure the lifted process does not explode before the
136300
original one does see (77], [34], and [43] p. 175. However we will not need
this.
B. For simplicity restrict attention now to an affine connection for M i.e.
a connection on GLM. Given a piecewise c1 curve
cs: [O,T)--. Manda vector v0 E T cs(o)M define the parallel translate
llt(v0 ) E Tcs(t)M 0 :s t :s T
of v0 along cs by
II 1(v 0 ) = cs-(t)b 0 -1(v 0 ) (14)
where cs-(t) is the horizontal lift of cs through b0 E n-1(cs(O)), the result being
independent of the choice of b0 •
Then lit: T cs(o)M--. T cs(t)M is a linear isomorphism.
If we now have vector field W along cs i.e. W : [O,T] --. TM with W(t) E
T cs(t)M for each t, define its covariant derivative along cs by
DW lot= lit dldt (//t-1 W(t)) (15)
Thus W is parallel alongcs i.e. W(t) =lit W(O) iff DWiot = o.
Over a chart (Uw~Pcx) forM, with induced trivialization of
n-1cucx), using the same notation as for (12)
T cs(t) ~Pcx (DWiot) = g(t) dldt (g(t)-1 v(t))
where v(t) = T cs(t) ~Pcx (W(t)), and so by (12) the local representative
T cs(t) ~Pcx (DWiot) is given by
dvldt + r(csa(t))(cra(t))(v(t)) (16)
for cra(t) = G>a(cr(t)) or, if olox1, ... ,oloxn denote the vector fields over Ucx
given by T x~Pcx (oloxi) = ei, the i-th element of the standard base of !Rn and if
W(x) = L.: wi(x) oloxi, and
DWiot = L.: (DWiot)i oloxi etc.,
then
(DWiot)i = dWildt + rijkCcscxCt))(cr(t)i)wk(t)
summing repeated indices.
(17)
By definition a curve cs in M is a geodesic if its velocity field cs is parallel
along cs i.e.
Dlot cr = o. Substitution of this into (17) gives the classical local equations. The existence
137301
theory for such equations shows that for each v0 e T x M there exists a unique 0
geodesic '¥0): 0 :5 t < t 0 for some t 0 > 0 with '¥(0) = x0 and y(O) = v0 • If we
can take t 0 = oo for all choices of v0 so geodesics can be extended for all time
the connection is said to be (geodesically) complete: (note that no metric is
involved so far). The geodesic y above is often written y(t) = expx tv 0 and 0
there is the exponential map defined on some domain .D of TM
exp : .D -+ M x M
exp v = (x, expxv)
when v e T xM· A use of the inverse function theorem shows that there is an
open neighbourhood .D 0 of the zero section Z[M] = image of Z : M-+ TM given
by Z(x) = 0 e T xM, such that exp maps .D 0 diffeomorphically onto an open
neighbourhood of the diagonal in M x M. In particular each expx : T xM-+ M is a
local diffeomorphism near the origin. The inverse determines a chart (U, cp)
around x by q> = expx-1: U-+ TxM ~ IRn. These are normal (or geodesic, or
exponential) coordinates about x. If '¥ is a geodesic in M from x then its local
representative in this chart, cp(y(t)) : 0 :5 t < t 0 , say, is just the 1/2 ray
segment (tv : 0 :5 t < t 0 , where v = 6(0). In particular we see from (15) and
(16) that for a torsion free connection, at the centre of normal coordinates
the Christoffel symbols (for that coordinate system) vanish.
C. Let IL(TM;TM) = U IL(T xM;T xM). It has a natural C00 manifold xeM
structure with charts induced by the charts of M , and a smooth projection
onto M , as do the other tensor bundles e.g. the cotangent bundle T*M = U IL(T xM;IR), the exterior bundles AP TM, and the bundles of p-linear maps
IL(TM, ••• ,TM;IR) ~ ®P T*M.
Note that a frame u at x determines an isomorphism
PI!... (u) ; IL(IRn;IRn)-+ IL(T xM; T xM)
PI!... (u)(T) = uTu-1
and similarly for the other buntjles mentioned:
p*(u): IRn*-+ T*xM given by p*(u)(.t) = ,tou-1,
and also
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and
P®(u)T = T(u-1(-), ... ,u-1(-)).
These equations also determine representations of GL(n) on O...(IRn,IRn),
A TM, etc. which will also be denoted by pn._, PI\• etc.
D. A vector field A on M determines a map
A~: GLM ~ !Rn
by
A-(u) = u-1 A(n(u))
Similarly a section B of O...(TM;TM), i.e. a map B:M ~ O...(TM;TM) such that B(x) E
O...(T xM;T xM) each x, gives
8~ : GLM ~ O...(!Rn; !Rn)
by
(18)
etc.
The covariant derivative VA of A is the section of O...(TM;TM) defined by
V A(x)(v) = u dA ~(v~) (19)
where v- is the horizontal lift Huv of v to HT uGLM, for v E T xM· This is often
written VA(v) or V vA.
Covariant derivatives of other tensor fields e.g. Sections B of O...(TM;TM)
are defined similarly: VB is the section of O...(TM;O...(TM;TM)) given by
VB(x)(v) = pn._(u) dB"'(v .... ) E O...(T xM;T xM) (20)
for v"' as before. In particular the higher order covariant derivatives are
defined this way, e.g.
v2A = V(VA)
is a section of
O...(TM;O...(TM;TM)) ~ O...(TM,TM;TM).
For a chart (Ucx, ~Pcx) forM around a point x, using the induced
trivialization of n-1cucx) our tensor field C, say, when lifted looks like a map
C' on Ucx X GL(n) given by c'(x,g) = p(g)-1 C'(x) where C' is c in our coordinate
system, and p is the relevant representation, e.g. p(g) = g for vector fields, p
= Pn... etc. In these coordinates v- = (v,- r(x)(v)) so VC(x)(v) is given by
dC'(v) + drp(f'(x)(v))C'(x) (21)
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where diP means the differential of p at the identity
-(r(x)v)*.
In particular if our vector field A is given over Uo: by
A= I Ai a;axi, etc.
then, summing repeated suffices, if Xo: = <po:(x)
[VA(v)]i = dAi(v) + rijk(Xo:)(vi)(Ak(x))
= (oAi;axD vi+ rijkCxo:)(vi)(Ak(x))
e.g. d1p*(r(x)(v) =
(22)
(where formally aAi;axi: Uo: -+ IR means the result of acting on Ai by the
vector field o/oxi : in practice everything is transported to the open set
<po:(Uo:) of IRn in order to do the computations so that oAi;axi is computed as
"o/oxj A i ( <po:(x 1, ••• ,xn))" in the sense of elementary calculus).
Comparing (17) and (22) one sees that if V is a vector field taking value v
at the point x, and if o is an integral curve of V, so cr(t) = V(O(t)), with o(O) = x then
DA/otlt=O = V A(v) = V vA (23)
Note that if V is a vector fleld we can form a new vector field VyA or
VA(V) by
V yA(x) := VA(V(x))
we see from (22), or by working at the centre of normal coordinates that for
a torsion free connection
(24)
D. Covariant differentiation behaves similarly to ordinary differentiation.
For example if o: is a 1-form (i.e. a section of T*M) and A is a vector field
ihen for v e T xM
d(o:(A(·)))(v)"' V vo:(A(x)) + o:(V vA(x)) (25)
One way to see this is to write o:(A(n(u))) = (o:x o u) o u-1A(n(u)) for
u e GLM, x = n(u). Then differentiate both sides in the direction Hu(v).
E. Using the notation of §3A of Chapter I
ct2/dt2 f(S(t,x)e) = d/dt df(Ye(S(t,x)e)) = d(df(Ye(-))(d/dt S(t,x)e)
which at t = 0
= V(df)(Ye(X),(Y9 (x)) + df(VYe(Ye(x)))
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by (25). Thus for any affine connection, (5) can be written
T
f(xT) = f(xs) + J df(X(xr)dzr) s
T
+ ~ J V(df)(Y(xr)dzr)(Y(xr)dzr) + '~f(VY(Y(x)dzr)dzr) (26)
s
which for equation (2)', gives the generator A in the form
Af = ~ ~ V(df)(XP(x))(XP(x)) + df(VXP(XP(x)))) + df(A(x)). p;l
(27)
Non-degeneracy of the S.D.S. (2') i.e. surjecti vi ty of each
X(x): IRm-+ T xM is equivalent to ellipticity of Af: the symbol of A is just
E XP ® xP as a section of TM ® TM, or X(·) oX(-)* as a section of
ll...(T*M;TM)).
§3. Riemannian metrics and the Laplace-Beltram1 operator
A. A Riemannian metric on M assigns an inner product < , >x to each tangent
space T xM of M, depending smoothly on x. Over a chart (Ucx:,<pcx) if
u = ui a;axi, v =vi a;axi are tangent vectors then define the n x n-matrix G(x)
= (gij(X)]i,j by
(28)
The inner product determines a metric d in the usual sense on M,
compatible with its topology, by letting d(x,y) be the infimum of the lengths of
all piecewise c1 curves from x toy, where the length .t(<5) is
b
.8(<5) = J lcr(t)l<5(t) dt a
1
for Julx = <u,u>2x as usual, and <5 is defined on [a,b]. The Riemannian manifold
(i.e. M together with<, >x: x eM) is (metrically) complete if it is complete
in this metric.
For a submanifold M of IRm the standard inner product of IRm restricts to
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an inner product < , >x on each T xM considered as a subset of IRm, thereby
determining a Riemannian structure on M. It is a highly non-trivial result, the
Nash embedding theorem, that for every Riemannian metric on a manifold M
there is an embedding into some IRm such that the induced metric agrees with
the given one i.e. an isometric embedding. (In general a smooth map f : M-+ N
of Riemannian manifolds is isometric if (T xf(u), T xf(v))f(x) = (u,v>x for all
x eM and u, veT xM; it is an isometry if it is also a diffeomorphism of M onto
N. Thus an isometric map need not preserve distance.)
B. Given such a metric one can consider orthonormal frames: these are
isomorphisms
u : IRn-+ T xM
preserving the inner products, (u(e),u(e')>x = (e,e'>IRn· The space OM of such
frames is a subset of GLM, and keeping n : OM-+ M to denote the projection it
forms a principal bundle with group O(n): it is a subbundle of GLM in the
obvious sense.
A connection on OM is called a Riemannian connection. w will take values
in the Lie algebra o(n) of O(n) which can be identified with the space of skew
symmetric n x n-matrices. It can be extended over all of GLM by the action of
GL(n) on GLM, insisting on condition (i) for a connection form (or (b) for the
corresponding horizontal subspaces). Thus it determines a connection on GLM
and so local coordinates have associated Christoffel symbols, which can be
used to compute covariant derivatives.
An important point is that for this induced connection on GLM, given a
curve <5 in M, the horizontal lift <5"" of <5 to GLM starting from an orthonormal
frame stays in OM and is the same as the horizontal lift for the original
connection on OM. An immediate consequence (from the definitions, equations
(14) and (15)) is that parallel translation preserves inner products:
(//tV• 1/tv')d(t) = (v,v')d(o) (29)
for v, v' e T <5(o)M, and for vector fields W, W' along <5
d/dt (W(t),W'(t))o(t) = (DW /at, W'(t))d(t) + (W(t), DW'/ot)o(t). (30)
Consequently, by (23), if W 1> w2 are vector fields and v e T xM then
d(W 1(-),W2(-))(-) (v) = (V W1(v), W2(x))x + (W 1(x), VW2(v))x (31)
C. The metric gives an 1 denti fi cation of T xM with its dual T x *M by
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v -+ v • = <v ,->x- In local coordinates (Uo:•<t'cx) let ~Po:(Y) = (x 1(y), ••. ,xn(y)) for y
E Uo: then dyx1, ... ,dyxn) form the dual basis to o/ox1, .•• ,o/oxn, (strictly
speaking eva 1 u ate d at y). If v = vi a 1 ax i at y then v• = vi d y xi where
vi = gij(y)vi (32) #
Write .2-+ 2 for the inverse of this isomorphism also.
By choosing the vector field A such that for given x E M and v e T xM, A(x) = v and \1 A(x) = 0, equation (25) shows that for a 1-form ex
(33)
Similarly \1 v commutes with the 'raising and lowering of indices' on other
tensor fields.
The gradient, grad f, or \lf, of a c1 function f: M-+ IR is the vector field #
(df) so
<Vf(x),v>x = df(v) (34)
all veT xM· In local coordinates \lf(x) = \lf(x)i a;axi where
\lf(x)i = gii(x) of ;axi (35)
where [gii(x)]i,j is the inverse matrix G(x)-1 to [gij(x)].
D. There will be many Riemannian connections for a given metric. However
it turns out that there is a unique one which is also torsion free. This is
called the Levi-Civita connection. It can be defined in terms of the
Christoffel symbols by
(36)
It is this connection which is usually refered to when considering
covariant derivatives etc. for Riemannian manifolds.
E. A Riemannian metric determines a measure on M, temporarily to be
denoted by Jl, such that if (Uo:, ~Po:) is a chart then the push forward Jlo: of JliUo:
by ~Po: is equivalent to Lebesgue measure on the open set <t'o:(Uo:) of IRn with
Jlo:(dx) = .[det G(<po: -1(x)) A(dx) where A is Lebesgue measure and G is the
local representative of the metric. We shall usually just write dx for Jl(dx) or
A ( dx) and write go:(x) or g(x) for det G(x) . Note -./g(x) = ldet T yiP ex - 11 for
y = ~Po:(x), where 'det' refers to the determinant obtained by using < , >x and
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< ' >IRn·
For a cl vector field A on M, the divergence, divA: M-+ IR, is given by
div A(x) = d/dt det T x Ft I (37) t=O
where
(t,x)-+ Ft(X) EM
is the solution flow of A, on its domain of definition in IR x M. It represents
the rate of change of volume by the flow. It is given by
div A(x) =trace VA(x). (38)
From (37), using the change of variable formula for Lebesgue measure one gets
the divergence theorem
J div A(x) dx = 0 (39) M
forM compact, and more generally. Since, by (22), iff: M-+ IR,
div fA(x) = (Vf(x),A(x)>x + f(x) div A(x) (40)
we see from this that div and -V are formal adjoints.
The Laplace-Beltrami operator fl. on c2 functions f: M-+ IRis defined by
M = div Vf
or equivalently
M =trace Vdf = l: V(df)(e;)(ei)
where e1, ... ,en are orthonormal. It determines a self-adjoint operator 6. on
L2(M;IR), [52], [91]. In local coordinates it has the formula
M(x) = gii(x) o2f;axioxi- gii(x) rkij(x) af;axk (41a)
and 1 . 1 . . .
M(x) = g(x)-2 o/oxl (g(x)2 gll(x) af ;axl (41b)
which are easily seen using (21) and (38) for (41a), and (35) and (37) for
(41b).
§4. Brownian motion on M and the stochastic development
A Let M be a Riemannian manifold with its Levi-Civita connection. By a
Brownian motion on M we mean a sample continuous process (Xt : 0 :s t < n, defined up to a stopping time, which is Markov with infinitesimal generator ~6..
From the Ito formula (27) a solution of
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dXt = X(Xt o dBt + A(Xt)dt
is a Brownian motion 1f and only if
(i) X(x) : IRm-+ T xM is a projection onto T xM for each x in M i.e. X(x) o X(x)* = identity; and
(ii) A(x) =- ~ L VXP(XP(x)) p
When (i) but not (ii) holds we say that (Xt : 0 :S t < s is a Brownian motion
with drift. The drift is the vector field x-+ A(x) + ~ L VXP(XP(x)). p
B. Note, again from (27) for an arbitrary affine connection, that in general
the generator A for our solution is elliptic if and only if each X(x) is
surjective (in which case the S.D.E. is said to be non-degenerate). In this case
each X(x) induces an inner product on T xM , the quotient inner product, and so
determines a Riemannian metric on M. Thus the solutions to a non-degenerate
S.D.E. are Brownian motions with drift for some (uniquely defined) metric on
M, and equivalently any elliptic A can be written as ~.1. + B for some
first order operator (i.e. vector field) B. Even working on IRn, if one wishes
to deal with elliptic generators A, the differential geometry of the associated
metric will not in general be trivial and can play an important role.
C. Although there always exist coefficients X and A satisfying (i) and (ii)
there is no natural choice which can be applied to general Riemannian
manifolds. However there is a canonical S.D.E. on the orthonormal frame
bundle OM toM, and it turns out that the solutions to this project down to give
Brownian motions on M. The construction, due to Eells and Elworthy, is as
follows:
Define X: OM x IRn-+ TOM by
X(u)e = Hu(u(e)). (42)
For given u0 in OM let (ut: 0 :S t < S be a maximal solution to
dut = X(ut) o dBt (43)
where (Bt: 0 :S t < oo is Brownian motion on IRn, so now m = n. Set x1 = TT(Ut):
Theorem 4C (xt : 0 :S t < S is a Brownian motion on M, defined up to its
ex~losion time.
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Proof Suppose g : M ~ IR is c2. Set f = g o TT : OM~ IR. With the notation of the
Ito formula (5), §3A, for e e IRn and u e OM set
'¥t(u,e) = n(S(t,u)e).
Then S(-,u)e is the horizontal lift of l_(u,e) through u. Since
'Yt(u,e) = Tn(X(S(t,u)e)(e))
= (S(t,u)e)(e) E T '¥t(U,e)M
(44)
'Yt is parallel along '¥t and sot -t 6t(u,e) is a geodesic in M, see §2B; also to=
u(e). Thus
and
d/dt f(S(t,u)e)lt=O = d/dt g(lt(u,e))lt=O
= dg(u(e))
ct2/dt2 f(S(t,u)e)lt=O = d/dt (dg(tt(u,e)))lt=O
= V dg(tt(U,e))lt=O + dg(D/ot 'Yt(U,e)lt=O)
= V dg(u(e))
since D/ot 'Yt = 0.
Thus by (5)
T T
g(xT) = g(x0 ) + J dg(ut o dBt) + ~ J 6g(xt)dt ( 45).
0 0
That Xt is a Brownian motion can now be deduced from the martingale
problem method, [92]. Alternatively, to show that Xt has the Markov property
first prove that the di stri buti ons Of Xt : 0 :$ t < s dO not depend on the point u0
e n-1(x0 ) in OM: having done this the Markov property is easily deduced from
that of the solutions of our S.D.E. on OM, (e.g. see [43], §5C of Chapter IX). To
see the lack of dependence up to distribution on u0 observe that if u'0 e n-1(x0 )
then there exists g e O(n) with u'0 = u0 .g. By the equivariance under O(n) of the
horizontal tangent spaces, condition (b) of §1C, ut·9: 0 :$ t <~)satisfies
du't = X(u'1) o dB't
where B't = g-1(Bt)· By the orthogonal invariance of the distributions of
Brownian motion, B't is again a Brownian motion and Ut·9 has the same
distributions as the solution u't of du't = X(u't) o dBt from u' 0 • Since TT(Ut.g) =
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n(ut) the required invariance for Xt follows.
The maximality of Cxt: 0 s t < ~) follows from that of Cut: 0 s t <~):if
Ut-+ oo in OM as t-+ ~(w) so does n(ut) since each n-1(x) is compact. II
D. Since each X(u)e E T uOM , the considerations of §2A suggest calling the
solution Cut: 0 s t < ~ the horizontal lift of the Brownian motion
Cxt:O s t <~from u0 : in fact it is easy to see that it locally satisfies the local
equations (12) considered as a Stratonovich equation with x1 replacing 6 1•
We can then define parallel translation along the sample paths of our
Brownian motion
llt(W): T x0
M-+ T Xt(w)M
by
WEQ (46).
Also if CWt: 0 s t <~)is a vector field along Cxt: 0 s t <~)i.e.
Wt(W) E T Xt(w)M WE Q
then its covariant derivative along the Brownian paths is the vector field along
Cxt: 0 s t <~)given by
DWtlot=llt dldt(ll1-1w1) (47)
E. We shall not give the details but any Brownian motion on M from a point x0
can be considered as obtained in the way described: as the stochastic
development of a Brownian motion on IRn. To do this we have to anti-develop
our given Brownian motion on M. A simple way to do this is to express Xt as
the solution of some S.D.E. dxt = Y(xt) o dzt, as described in §3C of Chapter I.
Take a horizontal lift Y~ of Y to OM so Y~(u)e = Hu(Y(n(u))e) for (u,e) E OM x
IRm. Solve
(48)
from a given u0 E n-1(x0 ). This is a candidate for a horizontal lift of
Cxt: 0 s t <~)(assuming it does live as long as Xt, which it does [34], [43]).
For the solution set
T
Bt = J Ut -1(Y(Xt) o dzt) E IRn
0
(49)
One then has to use the martingale characterization of Brownian motion to
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show that Bt is a Brownian motion (or at least part of one). It is rather
straightforward to see that Xt: 0 => t <~is its stochastic development.
Note: we have used Y so that (48) and (49) can be used rather than
discussing equations like the stochastic versions of (12): however, even for
general semi-martingales z, the above construction gives an 'anti-development'
with resultant !An-valued process (Bt in our case) which is independent of the
choice of Y. In fact the horizontal lift (ut : 0 => t => ~ given by (48) is
independent of Y as is most easily seen by the uniqueness of solutions to the
local equations (12). The horizontality can most easily be expressed by
T
J &(odut)=O
0
(50)
for any stopping time 0 => T <~where for an lAP-valued 1-form e:TN-+ IRP on
some manifold N, and a continuous semimartingale (Yt : 0 => t < ~) in N we
define
T T
J S(o dyt) = J e(Y(yt) o dzt) (51)
0 0
where T is a stopping time less than ~ and dYt = Y(y1) o dzt for some Y and z.
This has to be shown to be independent of the choice of Y and z: but this is
easily done either by working with local expressions or by embedding N in
some IRq and extending e and any suitable Y over IRq.
Given the horizontal lift, parallel translation along sample paths, and
covariant differentiation along the paths can be defined by (46) and (47). The
integral (51) of a 1-form eon M along the paths of Xt can be written
T T
J e (o dxt) = J e(ut o dBt) (52)
0 0
Note that this anti-development can be carried out for semi-martingales
on M given any affine connection on M, as can the development itself: however a
Riemannian connection was needed to construct Brownian motions since we
needed the invariance of the distributions of Bt under the group O(n). See [70],
[76] for other situations.
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F. The classical Cartan development maps a smooth path
<5(t): 0 ~ t < oo on IRn (or T x M), starting at the origin, to a path in M starting 0
at x0 • Mathematically it is just as described above with Bt replaced by <S(t) in
( 43) to yield the determi ni sti c equation
du/dt = X(u(t)) (d6/dt) (53)
with u(O) = u0 a given frame at x0 • The resulting path x(t) on M, defined for all
time, it turns out, if M is complete, is that which is obtained by the classical
mechanical procedure of "placing M on a copy of IRn (by u0 ), then rolling M
along <5(t) : 0 ~ t < oo without slipping and taking x(t) to be the point of
contact of M with IRn at time t". The frame u(t) : IRn-+ T x(t)M represents the
way M is resting on IRn at time t. When trying to visualize, or sketch, the
situation it becomes clear that the natural objects to use are affine frames, as
in [64], rather than the linear frames which we have used.
§5. Examples: Spheres and hyperbolic spaces
A. A connected Riemannian manifold is orientable if its orthonormal frame
bundle OM has two components. This is always true for M simply connected.
An orientation is then the choice of one of these, to be called SOM: it will be a
principal SO(n)-bundle whose elements can be called oriented frames. For
example there is a natural choice when M = sn given by the inclusion of the
tangent spaces T xsn into 1Rn+1 and the natural orientation of !Rn+1.
B. The natural left action of SO(n+1) on 1Rn+1 restricts to one on sn. If N
= (1,0, ••• ,0) is the 'north pole' of sn (with abuse of geography) the map of
SO(n+1) to sn
p g -+ g.N (54)
induces a diffeomorphism of the quotient space (with a natural differential
structure which we do not need to examine)
J3: SO(n+1)/SO(n)-+ sn.
In fact we will show that the oriented frame bundle sosn can be identified
with SO(n+1). However first we need some remarks about SO(n+1) itself:
The Lie algebra ll(n+1) will be identified with the space of skew
symmetric real matrices wlth .2Q(n) contained in it as
149313
~(n) = (g ~): 8 is n x n and B* = -B
There is then the vector space direct sum
s o(n+ 1) = so(n) + m
where
(0 -~t)
for s = ~ 0
with s written as a column.
Define
X : SO(n+ 1) x IRn -+ TSO(n+ 1)
by
(55)
(56)
so X(-)s is the left invariant vector field on SO(n+1) corresponding to the
element ! of m. Next define
by
ex: so(n+1)-+ sosn
ex(A)(s) = T A p(X(A)s)
= T 1(P o LA)!. (57)
To see this is an orthonormal frame it is only necessary to check that
T 1 (p o LA) : ill-+ T A.Nsn
is an isometry where m is given the inner product induced by s-+ f.. This is
easy to check when A is the identity, and follows for general A by noting that
SO(n+1) acts by isometries on sn (i.e. the derivative A*: T xsn-+ T A.xsn of x-+
Ax preserves the Riemannian metric for each x e sn). Similarly each ex(A) has
the correct orientation.
Finally observe that ex is equivariant for the right SO(n action, i.e. ex(Ag)
= ex(A)g:
In fact if Rg denotes right multiplication by g
poL A = (p o Rg -1) o LA = p o LA o Rg -1
forge SO(n), so
150
a(Ag)s = T 1 (p o LAg)t
= T1(P o LA Rg-1 Lg)t
= T 1(P o LA)(adg)-1t
= T 1 (p o LA)(gt)
314
(58)
(by an elementary computation). Thus p : SO(n+1)-+ sn has a principal bundle
structure isomorphic to 11: sosn-+ sn by (X.
There is a canonical connection on p: SO(n+1)-+ sn defined by
HA SO(n+1) = T 1 LA[m] A e SO(n+1) (59)
To see this is a connection note that forge SO(n)
T ARg[HA SO(n+1)] = T 1 LAg(adg)-1 [m]
= T 1 LAg(m) =HAg SO(n+1)
since m. is invariant under adg for g in SD(n). The connection form
w0 :TSO(n+1)-+ IRn is given by w0 (T 1LA(a + t)) =a for A e SO(n+1), a E ll(n)
and t em.
An important property is that it is invariant under the automorphism o0 of
SO(n+1) given by cs 0 = SAS where S = (-6 ~)
This connection induces a connection on n : sosn -+ sn via ex:, i.e. a
Riemannian connection on sn. In fact this is the Levi-Civita connection, a fact
which is proved using the invariance under cs 0 just mentioned: see [64] page
303.
C. This situation works much more generally. Another important example is
obtained by taking the lorentz group 0(1,n) of linear transformations of 1Rn+1
which preserve the quadratic form (Sx,x) for S as in §B above. This has 4
components: let G be its identity component,
G =A e 0(1,n): det A= 1 and A11 2: 1). (60)
Then the Lie algebra !J_ of G is just .Q. (n,1) where
Q(n,1) =A e O..(IRn+1;1Rn+1):Ats + stA = O. (61)
Let
M = x e 1Rn+1: (Sx,x> = -1 and x1 2: 1. (62)
Then the form v-+ (Sv,v) induces a Riemannian metric on M by restricting it to
each T xM and G acts on the left on M, preserving this structure with
151315
p:G-+M
given by p(A) = A.N for N = (1,0, ••• ,0), inducing a diffeomorphism of G/H with M
for
H =A e G : A.N = N = (~ ~):BE SO(n).
We can identify H with SO(n).
As before there is commutative diagram
a G SOM
~/ M
with ex equivariant under the right action of SO(n), so that p can be identified
with the oriented frame bundle n. For G there is the splitting
.Q.(1,n) = so(n) + m*
for
m* = (t* : s e !Rn, where t* = (~ ~J (63)
Again there is a connection ro 0 which induces the Levi-Civita connection for M,
see [64], p. 303, and an involution cs 0 •
The manifold M is n-dtmensional hyperbolic space Hn , see Exercise (ii)
at the end of §8 below and §4G of Chapter III.
The general situation where this works is that of a Riemannian
symmetric space see [64], [65].
D. In these situations the stochastic development construction is equivalent
to solving dgt = Xgt) o dBt for the X defined on G, (e.g. on SO(n+1) for sn), and
projecting by p to get Brownian motion on M. Thus for our solution with g0 = 1,
gt • N : t 2:: 0 is a Brownian motion on M starting at N.
Remark:
It is shown in [43] p. 257 that when M is a Riemannian symmetric space,
152316
so we can identify M with G/H as above, Brownian motions on G themselves
project to Brownian motions on M, so if g1 : t 2: o is a S.M. on G from 1 then
(gt.N:t 2: N is one on M starting at N, where N eM is arbitrary.
Similarly, for example by the discussion of the S(t,u)e in the proof of the
Ito formula, Theorem 4C, the geodesics of M from a point N are given by
((ExptA).N: t e !R where Exp is the exponential map of the group G, see §1A,
and A lies in a certain n-dimensional subspace of the Lie algebra of G : in
particular for M = sn we need only take A em, and for M = Hn we can take
A Em*.
§6. Left 1nvar1ant S.O.S. on Lte groups
A. For G a Lie group with left invariant Riemannian metric (i.e. each Lg is an
isometry) there is the left invariant system
dgt = X(gt) o dBt (64)
where n = m and X(1): !Rn-+ T 1 G is some isometry and X(g)e = Lg(X(1)e) fore e
!Rn and g E G. The solutions will be Brownian motions if L: VXP(XP(x)) = 0; p
in particular if each VXP(XP(x)) = 0. The latter means precisely that the
integral curves of d~/dt = XP(~t) are geodesics (just differentiate this equation
along its solution). For a Lie group with both left and right invariant metric
the i nversi ng map g-+ g -1 is an isometry so that if
)'(t): t E !R) is a geodesic so is y(t)-1: t E !R, and therefore )'(t)-1 = )'(-t). From this one can deduce that the geodesics are precisely the one-parameter
groups, i.e. the solutions of cr(t) = X(<l(t))e for some e (e.g. see [79]). Thus in
this bi-invariant case the solutions to (64) are Brownian motions.
The compact groups admit bi-invariant metrics, and conversely every G
with a bi-invariant metric is a product G' x !Rk with G' compact.
B. Given a bi-invariant metric on G we have just persuaded ourselves that
VA(A(x)) = 0 for all left invariant vector fields. Therefore if 8 and C are both
left invariant, by taking A = 8 + C we obtain, using (24),
VB(C(x)) = Hc,B](x) (65)
§7. The Second Fundamental form and gradient S.O.S. for an embedded
submani fold.
A. Suppose now that M is a submanifold of !Rn with induced Riemannian
metric. There is then a natural S.D.S. (X,B) on M where Bt: t 2: 0 is Brownian
153317
motion on IRm and X is just the orthogonal projection map P of §10, Chapter I,
(as in the S.O.E. of equation (7)) so X(x) : IRm -+ T xM is the orthogonal
projection.
Suppose f: M-+ IRis c1. Let f 0 : IRm-+ IR be some smooth extension. Using
V 0 for the gradient operator on functions on IRm we have df(v) = df 0 (v) for v e
T xM and so Vf(x) = X(x)(V 0 f 0 (x)) for x e M. Thus if <p : M -+ IR m denotes the
inclusion, writing <p(x) = (<p1(x), ••• ,<pm(x)) we see
XP(x) = VtpP(x) x e M,p = 1, ••• ,m. (66)
For this reason our S.O.S. is often called the gradient Brownian system for
the submanifold (or for the embedding <p). We will show that its solutions are
Brownian motions on M. For this we need ! VXP(XP(x)) = 0 all x eM, and so p
we will first examine how the covariant derivative for M is related to
differentiation in IRm.
B. Suppose Z is a vector field on M. Take some smooth extension which we
will write z0 : IRm -+ IRm. For vxM = (T xM)~ in IRm, as in §10 of Chapter I,
there is a symmetric billnear map
cx:x : T xM x T xM-+ vxM
called the second fundamental form of M at x , such that Gauss's formula
holds: for veT xM
OZ 0 (x)(v) = VZ(v) + CX:x(Z(x),v) (67)
One way to prove this is to define VZ(v) to be the tangential component of
OZ 0 (x)(v) and write Vx(Z 0 ,v) for its normal component. Then one can verify
that (Z,v) -+ VZ(v) satisfy the conditions which ensure it is the covariant
differentiation operator for the Levi-Civita connection on M and furthermore
show that Vx(Z 0 ,v) has the given form for a symmetric CX:x, e.g. see [65], pp.
10-13.
From this there is the bilinear map for each x in M:
Ax : T xM x VxM -+ T xM
defined by
<Ax(u,n,v> = <cx:x(u,v),S). (68)
If s:M-+ IRm is c1 with s(x) E VxM for all X E M and So is a c1 extension then
Weingarten's formula gives
154318
Ds 0 (x)(v) = -Ax(v,s(x)) +a normal component
In fact for x EM
(Z0 (x), s0 (x)) = 0
for Z and Z0 as before. Therefore if vET xM
(DZ0 (x)v, s0 (x)) + (Z0 (x), Ds 0 (x)v) = 0
i.e.
(69)
< o::x(Z 0 (x),v), s 0 (x)) + <Z 0 (x), tangential component of Ds 0 (x)v) = 0 proving
(69).
C. The following goes back to Ito's work published in 1950:
Proposltion 7C. The solutions of the gradient Brownian system for a
submanifold M of IRm are Brownian motions on M.
Proof For the constant vector fields EP(x) = (o1P, ••. ,omP), p = 1 tom on IRm,
EP(x) = X(x)EP(x) + Q(x)EP(x) for Q(x) = 1d- X(x). Therefore differentiating and
taking the tangential component, for veT xM
o = vxP(v)- Ax(v,Q(x)EP(x)). (70)
Thus if we choose our orthonormal base e1, ... ,em of IRm so that e1, ... ,en
are tangent to M at x
vxP(v) = o p = 1 to n (71)
while
vxP(xP(x)) = o p = n+1, ••. ,m
because XP(x) = 0 for such p.
Thus
k VXP(XP(x)) = 0 p
as required. I I
Note that if X0 : IRm-+ IL(IRm,IRm) extends X then the equation
dxt = X0 (Xt) o dBt
whose solutions lie on M when starting on M, has Ito form
dxt = X0 (xt)dBt + ~ k DX0 P(xt)(X 0 P)(xt))dt p
and for x EM
k DX0 P(x)(X 0 P(x)) = L O::x(X0 P(x), X0 P(x)) = trace O::x p p
by (72) and (67). The standard example of this is the equation
(71)'
(72)
155319
dxt = dBt- <xt,dBt> lxtl-2xt- Hm-1)lxtr2 x1dt (73)
which gives Brownian motion on the sphere sm-1(r) of radius r if
x0 E sm-1(r). For variations, extensions, and further references consult
[95].
Note that from (70)
di v XP(x) == trace Ax( -,Q(x)EP(x)) L~= 1 ( a:x(Eq(x), Eq(x) ), Q(x)EP(x))
= L~=l (trace a:x, Q(x)EP(x)) (74)
D. The mean curvature normal at x is 1/n trace a:x E VxM. For a
hypersurface vxM is a 1-dimensional subspace of IRm and a choice of
orientation in vM (e.g. outward normal) gives the mean curvature as a real
valued function on M.
Similarly for a hypersurface the second fundamental form can be treated
as a real valued bilinear form on the tangent spaces to M. At a point x its
eigenvalues are called the principal curvatures at x, and its eigenfunctions
are the principal directions at x.
§8. Curvature and the derivat1ve flow
A. Given intervals I and J of IR and a piecewise c1 map u: I x J-+ M, there are
vector fields ou/os and ou/ot over U (i.e. the derivatives with respect to the
first and second variables respectively). Taking normal coordinates centred
at a point u(s,t) it is immediate from (17) that
D/ot au;as = D/os au;at (75)
If F t : M x Q-+ M, t 2: 0, is a smooth flow for our S.O.S. and v E T xM, choose
o: [-1,1]-+ IR with 6(0) = x and cr(O) = v. Then
T xFt(v) = a;as Ft(CI(s)) a.s. (76)
Defining parallel translation via a horizontal lift as in §§4D, E, we can
covariantly differentiate (76) in t using the analogue of (47) as definition, to
get
Dvt = Do/os Ft(CI(s))
for Vt = T x F1(v) : Q-+ T x1M, and so by the analogue of (75), proved in exactly
the same way,
Dvt = 0/oS Y(Ft(CIs)) • dZt
156320
i.e.
(77)
For later use, and as an exercise, we will find an equation for lv1F assuming now that we have a Riemannian metric and are using its Levi-Civita
connection. Certainly there is the Stratonovich equation
dlvtl2 = 2(vt, o Dvt>xt = 2<vt, VY(vt) o dzt>xt (78)
obtained, for example, by parallel translation back to x. A safe way to get the
Ito form is to use the Ito formulae (5): let S(t,x)(e) denote the flow of Y(-)(e)
and set &S(t,v) = T x(S(t,x)(e))(v) for veT xM, so
D/ot &S(t,v) = VY(&S(t,v))(e).
Therefore
d/dt !&S(t,v)l2 = 2(&S(t,v), VY(&S(t,v))e>
= 2(v, VY(v)e> at t = o and, at t = 0,
d2/dt2 I &S(t,v)j2 = 2(VY(v)e, VY(v)e) + 2(v,VY(VY(v)e)e)
+ 2(v,v2(Y(x)e,v)e) (79)
where, for Z a vector field and u e T xM,
v2z(u,-) = v u(VZ) : T xM + T xM (80)
Thus
dlvtl2 = 2(Vt,VY(vt)dzt> + (VY(vt)dZt,VY(vt)dzt>
+ <vt, VY(VY(vt)dzt)dzt) + <vt,v2Y(Y(x1)dzt,vt)dzt> (81)
(To be convinced of the applicability of equation (5) observe that Vt is actually
a solution of the S.D.E. on TM, dvt = &Y(vt) o dzt, given in §4E, Chapter I.) We
shall simplify (81) in §8C below.
B. The curvature R of an affine connection is a section of the bundle IL(TM,
TM; IL(TM;TM)) so it can be considered as a map
R : TM ffi TM -+ IL(TM;TM)
where TM ffi TM = U(T xM ffi T xM : x e M), such that
u,v,w-+ R(u,v)w
is tri-linear in u,v,w e T xM· It can be defined by
R(u,v)w = v2w(u,v) - v2W(v,u) (82)
where W is a vector field such that W(x) = w at the given point x of M. We will
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see below in §9C that this definition gives a result independent of the choice of
such W. If U, V, Ware all vector fields there is the new one R(U,V)W given by
x -+ R(U(x), V(x))W(x)
so
R(U,V)W = Vu Vv W- Vv VuW- V[u,v]W (83)
for a torsion free connection using (24) and §1D, Chapter II).
From 82) R is anti-symmetric in its first two variables. For the Levi
Civita connection of a Riemannian metric it turns out that (R(u1,v1)u2,v2>x is
antisymmetric in u1, v1 and in u2, v2 and satisfies
(R(u 1,v1)u2,v2>x = (R(u2,v2)u1,v1>x 84)
This is an automatic consequence of the skew-symmetry and the relation
R(u,v)w + R(v,w)u + R(w,u)v = 0 (85)
which can be proved by choosing vector fields U, V, W with U(x) = u, V(x) = v,
W(x) =wand which commute i.e. [U,V] = [V,W] = [W,U] = 0. For details see e.g.
[64] or [79] (where a different sign is used!).
For the Levi-Civita connection the Riemannian curvature tensor is defined
by R(v1, v2, v3, v4) = (R(v1, v2)v4, v3>x
=- (R(v1,v2)v3,v4>x (86).
for vi e T xM, (note the sign difference from [33]) and the sectional curvature
Kp(x) of a plane P in T xM is
Kp(x) = R(vv v2, v1, v2) (87)
where v 1, v2 is an orthonormal base for P.
When M is a surface, i.e. n = 2, there is a unique P at x, namely P = T xM,
and then Kp(x) coincides with the classical Gaussian curvature of M at x. For
M a submanifold of IR3 wHh induced metric it was Gauss's famous theorem
aegregium which showed
Kpx) = :>.. 1 :>..2 (87)
where :>..v :>.. 2 are the principal curvatures at x §7D), see [65] for example.
Continuing with the Riemannian case there is the Ricci curvature
Ric : TN EB TN -+ IR
Ric(v 1,v2) =trace [v-+ Rv,v 1)v2J = L. (R(ei,v 1)v2,ei>
J
(88)
for e1, ... ,en an o.n. base in T xM· Clearly it is symmetric. Observe that Ric(v,v)
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is the sum of the sectional curvatures of any family of (n-1) mutually
orthogonal planes P in T xM containing v, provided lvl = 1. Thus bounds on the
Ricci curvature are weaker assumptions than bounds on the sectional
curvatures.
The Ricci curvature plays a very important role in the study of diffusions.
An important result of differential geometry is that if the Ricci curvature is
bounded below i.e. if there exists a constant C with
Ric(u,u) 2: Clul2 for all u e TM
then for any x0 E M if r(x) = d(x,x 0 ) then t::,r is bounded above uniformly at
all points where r is differentiable [101], [58]. From the Ito formula
applied tor as if it were c2 the result of 5.-T. Yau that a complete
Riemannian manifold with Ricci curvature bounded below is stochastically
complete (i.e. its Brownian motion does not explode), is no surprise [43], p.
242: see [38] for an analytic proof, and [ 61] for a probabilistic version with
an Ito formula for r(xt) involving a local time at the points where it is not
differentiable.
Taking the trace of the Ricci curvature at x gives the scalar curvature, a
real valued function K : M-+ IR
Note that for a surface K(x) is twice the Gaussian curvature
K(x) = Ric(e 1,e 1) + Ric(e2 ,e 2) = 2Ric(eve1)
(89)
= 2Kp(X) (90)
A space is said to have constant curvature if all its sectional curvatures
Kp(x) are the same (it suffices to know they are independent of P for each
x EM by a result of Schur, see [64]). If so
R(u,v)w = k ((w,v)u- (w,u)v) (91)
where k = Kp(x). For such a space of constant curvature k we see
Ric(u,v) = (n-1)k (u,v>x (92)
and
K(x) = n(n-1)k (93)
C. We can now get improved formulae for the norms of the derivative flow.
Suppose the original S.D.E. was dxt = X(x1) o dBt + A(xt)dt with generator A=
159323
~t. + Z, where Z is a vector field and M is Riemannian. Write A= W + z. Then,
for each x e M, by §4A
~ L VXP(XP(x)) + W(x) = 0 p
Differentiating this in the direction of v, for veT x M
Lp v2 XP(v,XP(x)) + Lp VXP(VXP(v)) + 2VW(v) = 0
whence, by (82) and the definition (88) of Ric
2: (V2XP(XP(x),v),v) + Ric(v,v) + (2: VXP(VXP(v)) + 2VW(v),v) = 0 (94) p p
Substituting in (81):
dlv1!2 = 2(Vt,VX(v1)dBt) + 2(Vt,VZ(Vt))dt
- Ric(vt,Vt)dt + :l:p (VXP(v1),VXP(vt))dt (95)
In particular we see lv11 e L2(Q,~,IP) provided the quadratic forms
v-+ 2(v,VZ(v)) -Ric (v,v) and
v-+ IVXP(v)l2, p = 1,2, ••• ,m
for v e T xM are bounded above uniformly over M. Equation (95) and the growth
of lvtl will be examined in detail in special cases in Chapter III.
§9. Curvature and torsion forms
A. When trying to find a useful analogue of (~lS) for the canonical S.D.S. on
the frame bundle of a Riemannian manifold we shall need the curvature form,
so we will describe it here and use it to prove some of the basic results about
curvature which were stated in §8.
First suppose we have a principal G-bundle n : B -+ M with a connection
form & • This is a g-valued 1-form on B. As with a real valued one-form it
has exterior derivative
d& : TB ffi TB -+ g
(v 1,v2)-+ d&(v 1,v2)
which is antisymmetric, satisfying
dw(U(b),V(b)) = U(&(V))(b)- V(&(U))(b)- &([U,V](b)) (96)
for U,V vector fields, where &(V) is the g-valued function on B, b -+ &(V(b)),
etc; see the discussion of differential forms in Chapter IV.
N.B. Here we are departing from the convention of Kobayashi and Nomizu
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[64], which would have a factor of~ multiplying the right hand side of (96).
The curvature form is the 2-form R given by
R : TB EB TB -+ g
(97)
for v1, v2 e TbB where h : TB -+ HTB is the projection onto the horizontal
subspace. By the invariance property (B) of w, i.e. w o TRg = ad(g-1) ow , we
have
(98)
for b e 8, g e G, and v1 , v2 e TbB, because of the invariance of exterior
differentiation under diffeomorphisms: in our case
d(ro o TRg)(v 1,v2) = dro(TRg(v 1),TRg(v2)).
B. To see the importance of R suppose v1, v2 are horizontal vector fields on
B so Vi(b) e HTbB for each b. Then ro(Vi(b)) = o for all band so by (96):
R(V 1(b), V2(b)) =- ro([V 1,V2 ](b)) (99)
Thus if R. vanishes identically the Lie bracket of horizontal vectors is
horizontal: but this is precisely the classical necessary and sufficient
condition of Frobenius' theorem for the integrability of (HTbB: be B) i.e. for
the existence of a submanifold through each b of B with HTbB as its tangent
space. (In particular if this happens for a Levi-Civita connection the canonical
S.D.S. will be far from hypo-elliptic).
C. Now suppose we have an affine connection, so B = GLM, or OM for the
Riemannian case. Then there is the canonical1-form
e : TB -+ !Rn
e(v) = u-1(TTT(v)) veT uB
From this we obtain the torsion form
®(v 1,v2) = d9(hv 1,hv2)
It satisfies
8(TRgv1, TRgv2) = g-1 ®(v1,v2)
Because of (98) and (102) there exist
T : TM EB TM -+ TM
bilinear and skew symmetric from T xM EB T xM-+ T xM and
R : TM EB TM -+ IL(TM;TM)
again bilinear and skew symmetric, defined by
(100)
(101)
(102)
161325
(103)
and
R(v1,v2)v3 = u R (Huv1, Huv2)u-1(v3) (104)
for v; e T xM, u e n-1(x), and Hu the horizontal lift operator.
These are the torsion and curvature tensors of our connection. We must
show that this definition of R coincides with the one in §8:
Proposition 9C.
ForT, R defined by (104), if V,W are vector fields on M and v1 ,v 2
tangent vectors at x toM
( i) If T = 0 then
v2v(v1,v2)- v2v(v2,v1) = R(v1,v2)V(x)
(ii) [V,W](x) = VW(V(x))- VV(W(x))- T(V(x),W(x)) (104a)
In particular the connection is torsion free iff T = 0 or equivalently® = 0.
~
(i) Choose u0 e n-1(x) and define the vector field V; on 8 by
V;(u) = Hu(uu0 -1v;) i = 1,2 ..
If T = o then
0 = ®(V 1,v2) = d9(V 1,V2)
= v 1 ecv2)-v2e(v 1) - e([v 1,v2])
= - e([v 1.V2D
since e(V;) is constant for each i. Thus Tn([VvV 2 ]) = o and so [v 1,v 2 ] is
vertical. It follows that
[V 1,v2](u0 ) = A*(u0 )
for
A= ro([V1,V2](u 0 )) =- R(V1(x), V2(x))
(by equation (9) and (99)).
Now by the definitlon, (19), for v(u) = u-1v(n(u)), etc.
V2V(V 1•V2) = u0 d(Vv)(V 1 (u0 ))(u0 -1v2)
= Uo V1(Vv(-)(ua-1v2))
= u0 V 1V2(V)(u0 ) (105)
Thus V2V(v1,v2)- v2v(v2,v1) = u0 [V 1•V2JV(uo)
= u0 A* v(u0 )
162
as required.
326
= u0 d/dt V(u0 exptA) I t=O
= u0 d/dt exp(- tA)u 0 -1v(x) I t=O
= -u0 Au0 -1 V(x)
= u0 R(v1,v 2)u 0 -1v(x)
Part (ii) can be proved similarly. II
D. Example
Let us compute the curvature of the sphere sn using the identification of
sosn with SO(n+1) as in §5. Using the notation of §5 to computeR., because of
left invariance it is enough to compute it at the point 1. For this let S• 11 E !Rn
with corresponding t and n. in m.. Take the left invariant vector fields U, V on
SO(n+ 1) which are t and n. at 1. Then
[U,V](1) = [t, .nJ =a e ~(n) which confirms that [U,V] is vertical (cf. the proof of Proposition 9C).
Therefore, by definition of the connection form ro 0 and equation (99), if Q is
the curvature form
R(t, n.) = - .e.. Now if v E !Rn, (identified with (O,v) in 1Rn+1) we see
rt.n.Jv = -<1l.v>s + <s.v>'Tl
Also T 1 TI: T 1 SO(n+1)-+ TNsn c!Rn+1 is just
A -+ ( O,cx:)
(106)
where ex: is the component of A in m., since TI maps g to g.N. Thus, if vi e TNSn
= 0) x !Rn c 1Rn+1, fori = 1,2,3, the horizontal lift H1(vi) = Y.i and
R(v1,v2)v3 = R(Y.i,Y.i)v3
= <v2,v3)v 1 - <v 1•v3>v2
which shows that sn has constant curvature+ 1, (c.f. equation (91)).
Exercises
(i) Check that the torsion form ® vanishes identically, so that we have a
complete proof that ro 0 gives the Levi-Civita connection.
(ii) Do the same for hyperbolic space Hn, showing it has constant curvature -
1; (the difference is in the use oft* given by (63) instead of tl
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110. The derivat1ve of the canonical flow
A. Let M be a Riemannian manifold with Levi-Civita connection. Suppose, for
simplicity of notation, that its canonical S.O.S. (X,B) on OM is strongly complete
with flow Ft(u,w): t 2:: 0, u e OM, we Q). To get equations like those of §SA for
TuFt we would need an affine connection for OM rather than just M. There are
various candidates, e.g. see [77], but the computations get rather complicated
and we will go by a direct method as in [43] (which in fact boils down to using
the 'canonical flat connection' for OM).
The derivative flow lives on TOM. However there is a canonical
trivializatlon of this tangent bundle: i.e. a diffeomorphism
\j) : TOM -+ M x !Rn x Q(n)
which restricts to a linear isomorphism \j)u: T uOM-+ u x IRn x Q(n) for each
u e OM. This is given for V e T uOM by
\j)(V) = (u,9(V), ro(V)) (107)
fore, ro the fundamental form and connection form. Using this, for V0 e T u OM 0
we will describe TF 1(V 0 ) in terms of !Rn- and Q.(n)-valued processes st =
9(TFt(V 0 )), At= ro(TFt(V 0 )). This is similar to a procedure used in the
Malliavin calculus [17], (although here we have differentiation with respect to
the initial point, and there a 'differentiation' with respect to the basic noise
Bt: t 2:: 0 is being considered as in [42a]) ••
We will use the analogous notation to §SA so S(t,u)e gives the flow on OM
of ut = X(u1)(e) fore e !Rn, and &S(t,V) gives its derivative flow on TOM.
B. A vector field J along a geodesic 1 in M is a Jacobi field if it satisfies
o2J;at2 + R(J, dl/dt) d"t/dt = o (10S)
They arise as infinitesimal variations of geodesics, [79], (see the proof
below) and as a method of computing derivatives of the exponential map.
Lemma 108. For V e T uOM and fixed e in !Rn set J(t,V) = Tn(&S(t,V)). Then
J( -, V) is a Jacobi field with
J(O,V) = Tn(V)
D/ot J(t, V)lt=O = u ro(V)(e). (109)
Proof First recall from §4C, equation (44), that t-+ n(S(t,u)e) = lt(u,e) is a
164328
geodesic: so J(-,V) is a vector field along a geodesic. Next take a horizontal
path o in OM with o(O) = u and 0"(0) = hV, and the path gin O(n) given by
g(s) = exp s ro(V).
Set p(s) = o(s).g(s). Then p(O) = V
so that
J(t,V) = olos (nS(t,p(s))e)]s=O
= olos 6't(p(s),e)ls=o·
Therefore
Dlot J(t,V) = Dlos a1at lt(p(s),e)ls=O
and by (82)
o21at2 J(t,V) = Dlos Dlot 'Yt(p(s)),e)ls=O
-R(J(t,V), y1(u,e)) y1(u,e)
(110)
which shows J is a Jacobi field since Dlot olot lt(p(s),e) = o because l is a
geodesic.
Clearly J(O,V) = Tn(V) as claimed, and also by (110) and the proof of
Theorem 4C:
Dlot Jlt=O = Dlos p(s)els=O = Dlos o(s).g(s)els=O
= o(s) dlds g(s)els=O = u ro(V)e. 1 I
C. From the lemma we have
dldt e(oS(t,V)) = dldt (S(t,u)e)-1 Tn(oS(t,V))
= (S(t,u)e)-1 Dlot J(t,V)
= ro(V)e
at t = 0, and, at t = 0,
ct21dt2 e(oS(t,V)) = u-1 o21at2 J(t,V)
= -u-1 R(TnV,ue)ue
(111)
(112)
Now take any affine connection for the manifold OM which is torsion free
(e.g. the Levi-connection for the metric on OM induced by t.p ). For this
dldt ro (oS(t,V)) = Vro(X(S(t,u)e)e)(oS(t,V)) + ro(DiotoS(t,V)) (113)
However, by §SA,
Dlot oS(t,V) = VX(oS(t,V))e
and also since X(u)e is horizontal, ro(X(u)e) = 0 for all u e OM whence
Vro(V)(X(u)e) + ro(VX(V)e) = 0 (114)
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for all VeT uOM. Substituting in (113)
dldt &(&S(t,V)) = V&(X(S(t,u)e)e)(&S(t,V))
- V ro(&S(t, V))(X(S(t,u)e)e)
= d&(X(S(t,u)e)e, &S(t,V))
by (96) (choosing suitable U,V which commute at S(t,u)e).
(115)
To proceed further we need the first of the following, and we state the
second in passing; they are valld for any affine connection, and the first for
any connection on a principal bundle:
Structure Equations: For V1,V2 e T uOM
(i) d&(V 1,v2) =- [&(V 1), &(V2]) + R(V 1,V2)
(ii) d8(V 1,V2) =- (&(V 1)e(V2)- &(V2)e(V 1)) + 0(V 1,v2) (117)
Proof We prove only (i), for (ii) see [64] Theorem 2.4, Chapter III, (the ~·s in
[64] come from the different convention for exterior differentiation used
there).
For (i): if both V 1, v2 are horizontal the result is clear by definition of R.
If both are vertical we can suppose V 1 = A *(u), v2 = B*(u) at the given point u,
for A, B e Q(n). Then
d&(A*,B*) = A*&(B*)- B*ro(A*)- ro([A*,B*J)
= - [A,B] = - [w(A *), w(B*)]
since ro(B*) and w(A*) are constant and [A*,B*] = [A,B]*. This gives (i) in
this case because R(A*,B*) = 0.
To complete the proof it is enough to suppose v1 is horizontal, v1 = V(u)
for some horizontal vector field V, say, and v2 "' B*(u) some Be Q(n). Again
d&(V 1,V2) = V ro(B*(u))- B*w(V 1)- ro([V,B*](u))
= - ro([V ,B*](u)).
Thus (i) is equivalent to [V,B*] being horizontal. However this is clear since
[V,B*](u) = dldt TWt(V(u.exp(-tB)))!t=O where Wtu = u.exp(tB) and horizontality
is preserved under right translation. I I
Applying the first structure equation to ( 115):
dldt &(&S(t,V)) = R(X(S(t,u)e)e, &S(t,V)) (118)
= (S(t,u)e)-1R((S(t,u)e)e, J(t,V))S(t,u)e (119)
and so, at t = 0
166330
d2/dt2 ro(&S(t,V)) = u-1 D/ot R((S(t,u)e)e, J(t,V))It=O u
= u-1 VR(ue)(ue, ue(V))u
+ u-1 R(ue, u ro(V)e)u
since D/t (S(t,u)e)e = 0.
(120)
From these we have our equations, in Stratonovich form by (111) and
(119):
dst = At o dBt
dAt = Ut-1 R(Ut 0 dBt, Ut St)Ut
(121a)
(121b)
and in Ito form using (112) and (120), for e1, .•• ,en an orthonormal base for IRn:
dst =At dBt- ~ Ut-1 Ric(ut svt (122a)
dAt = u1-1 R(ut dBt,Utst)Ut + ~ (ut-1 Z:i VR(utei)(Utei,utst)Ut
(122b)
(It is shown in [43] p. 168 that the covariant derivative of R can be replaced by
a term in the covariant derivative of the Ricci tensor since for an orthonormal
base f 1, ... ,f n for T xM
((Z:i VR(fi)(fi,w))v 1,v 2>x = VRic(v2)(v 1,w)- VRic(v 1)(v 2,w) (123)
for all v1,v2, win T xM.)
Note that if dBt is replaced by dt in (121a,b) we obtain the Jacobi field
equation for ut St· The fields Tn(TF1(V)): t 2:: 0 along the Brownian motion
n Ft(u): t 2:: 0) have been called stochastic Jacobi fields, [71].
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CHAPTER III: CHARACTERISTIC EXPONENTS FOR STOCHASTIC FLOWS
§.1. The Lyapunov Spectrum
A. Suppose throughout this section that M is a compact connected Riemannian
manifold with smooth S.D.S.
dx1 = X(xt) o dBt + A(xt)dt
differential generator A. Let Ft: M x Q-+ M, t;:: 0 denote its flow so for each
w E Q we have a C00 diffeomorphism Ft(-): M-+ M, derivative TFt(-,w) : TM-+
TM. Let (Q,~,!P) be the classical Wiener space of paths starting at 0 in lRm
with Bt(w) = w(t), and let 9t: Q-+ Q be the shift:
9t(w)(s) = w(t+s)-w(t) (124)
Then lP is invariant under e1 fort;:: 0.
A Borel probability measure p on M is invariant for our S.D.S. if
IE p o Ft(-,w)-1 = p t;:: 0 (125)
Since M is compact there exists an invariant measure (e.g. see [102], XIII
§4). The invariance of p depends only on A, not on the choice of S.O.S. with .A.
as generator. When A is elliptic then pis a smooth measure i.e. p(dx) = :>..(x)dx
for some smooth A. where dx refers to the Riemannian volume element: it is
also unique. This is because :>.. is a solution to the adjoint operator equation
(e.g. see [59]).
Define
'Pt : M X Q -+ M X Q
by
'Pt(x,w) = CFt(x,w), e1w).
Then for each s,t;:: 0
'Pt 'Ps = 'Pt+s a.s.
since
Ft<Fs(x,w), esw) = Fs+t(x,w) a.s.
Also if p is invariant for the S.D.S., then p 0 lP is invariant for 'Pt since if
f: M x Q-+ lR is integrable
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Jf f(Ft(x,w),e1w)p(dx) IP(dw)
= f Jf f(Ft(X,W1), 8tW2)p(dx) 1P(dw1)1P(dw2)
(because e1w is independent of Ft(x,w))
= If f(x,w) p(dx) IP(dw).
Say that p is ergodic if p 0 IP is ergodic for ~t : t 2: 0 i.e. if the only
measurable sets in M x Q which are invariant under ~ 1 : t 2: 0 have p 0 IP
measure 1 or 0. This agrees with the definition in [102]. An ergodic
decomposition for any invariant pis given in [102].
B. In this chapter we shall be mainly concerned with looking at special
examples of the following version by Carverhi1l [20] of Ruelle's ergodic theory
of dynamical systems:
Theorem 18 Let p be an invariant probability measure for A • Then there
is a set r c M x Q of full p 0 IP-measure such that for each
(x,w) E r there exist numbers
:>-_(r)(x) < .•• < :>-_(1)(x)
and an associated filtration by linear subspaces ofT xM
0 = v(r+1)(x,w) c v(r)(x,w) c ... c v(1)(x,w) = T xM
such that if
then
V E V(j) (X,W) - V(j+1)(X,W)
lim 1/t log IITFt(v,w)IJ = :>-_(j)(x)
t-+oo
(126)
where II II denotes the norm ustng the Riemannian metric of M. Moreover
for (x,w) E r the multiplicities mi(x) := dim v<Dcx,w) - dim v(i+1)(x,w) do not
depend on w and if
(127)
then
:>-.L:(x) =lim 1/t log det T xFt(-,w). (128)
(strictly speaking we should write ldet T xFt(-,w)l here and below or use some
other convention to ensure it is continuous in t with value 1 at t = 0) •
Proof
Following [20] embed M in IRn+p for some p and extend X,A just as in §2C of
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Chapter I to give X, A on IRn+p with compact support. Let Ft(x,w) refer to the
flow of this system, and <l)t to the flow on !Rn+p x Q; p remains an invariant
measure on IRn+p, concentrated on M.
Fix some time T > 0 and for n = 0,1,2, ••• set
Gn(x,w) = DFT(<l>nT(x,w)): IRn+p-+ !Rn+p
(where the D refers to differentiation in IRn+P). Set
Gn(x,w) = Gn_ 1(x,w) o ••• o G0 (x,w)
so
Gn(x,w) = DFnT(x,w),
by the chain rule.
and
Fairly standard estimates show that both
log+ IIDFT(x,w)ll e L 1(M x Q; p ® IP) (129)
(130)
We can therefore apply the Oseledec multiplicative ergodic theorem, as in
00 Ruelle [87], to (Gn(x,w) n=O and obtain the theorem in a discrete time version,
for X, A, and with the :Ai and mi possibly depending on (x,w).
To deduce the continuous time version from this as in [87] the
integrability of sup log IIDF t(x,w )II and of
O~t~T
sup log I!D(FT(-,w) o F1(-,w)-1)Ft(x,w)ll
O~t~T
are used. Having done this the filtrations for Ft on M are obtained by
intersecting those for F with each T xM. To show that the :Ai and mj do not
depend on w as described, [24], use the fact that this is certainly true if p is
ergodic (in which case they can be taken independent of x for suitable choice of
r), and then the fact that, even if not ergodic, p can be decomposed into ergodic
measures concentrated on disjoint subsets of M, [102]. I I
C. When the system is non-degenerate p is ergodic, and unique, and r can be
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chosen so that the exponents Ai and multiplicities mi, and consequently the
mean exponent 11n AL;, are all independent of x.
D. When p is ergodic and an exponent Ai is negative there is a stable
manifold theorem due to Carverhill [20]. Its proof comes from Ruelle's in
[87] for deterministic systems using the embedding method and regularity
estimate, for T > 0,
IE sup IIF t (-, w) II 2
< oo o:;:;~ c
Theorem 10 (Stable manifold theorem) For ergodic p with A(j) < 0 for
some j the set r of Theorem 18 can be chosen so that for each (x,w) E r the set y(j)(x,w) given by
y(j)(x,w) = y: lim sup 11t log d(F t<x.w ),F 1(y,w)) ~ A (j)) ( 131) t~ 00
is the image of v(i)(x,w) under a smooth immersion tangent to the identity at
X. I I
This means there is a smooth f : v(i)(x,w) ~ M which has T vf injective for
each v in v(i)(x,w) and has T 0 f = 1d, with yj)(x,w) as its image. It will be
locally a diffeomorphism onto its image but not necessarily globally.
If A (1
) < 0 we say the system is stable. In this case y(1)(x,w) will be an
open subset of M.
N.B. When M is compact any two Riemannian metrics ((-,->x: x EM
(-,->'x: x EM are equivalent: there exists c > 0 with c-1(u,u>'x ~ (u,u>x ~
c(u,u)'x for u e: TxM· Consequently the Lyapunov spectrum and stable
manifolds are independent of the choice of metrics. This would not be true if
we allowed M to be non-compact e.g. as in [26], [27).
E. The first examples to be studied were the 'noisy North-South flow on s1·
(i.e. in stereographic projection so the North pole corresponds to oo and the
South pole to 0 the equation is
dXt = e: Y(Xt) o dBt - dt
where (Bt: t 2: 0) is one-dimensional andY corresponds to the unit vector field
on s1), with variations [21], and gradient Brownian flows, canonical flows on
the frame bundle, and some stochastic mechanical flows, [22]. The latter are
171335
usually flows on the non-compact space IRn of the form
dXt = dBt + V log \j) dt (132)
where \j) is a smooth l2 function, with V\j) in l2, which is positive (at least for
the 'ground state' flow when it is the leading eigenfunction of Schrodinger
operator-~ t::. + V, for V a real valued function). However the estimates still
work to allow the exponents to be defined, [26], [27]; and something can be
said even in some cases when \j) is time dependent (the quasi-periodic case, for
lj) 0 a linear combination of eigenfunctions of our operator). For spaces of
constant curvature, or more generally for Riemannian symmetric spaces, the
frame bundle can be identified with p: G-+ M:::: G/H for some lie group Gas in
§5 of Chapter II, and the study of the asymptotic behaviour of the canonical
flow in this situation was begun by Malliavin and Malliavin in 1974!75 [71],
[73]. We shall look at gradient flows and canonical flows, and for the latter
look in detail at the situation for the hyperbolic plane. However we will
mainly look at it directly rather than use the gr·oup theoretic approach because
the latter is special to the case of symmetric spaces and so not likely to be
much help in obtaining results for spaces with non-constant curvature. The
group theoretic approach is carried through in detail in [10], where there is
also a very nice analysis of gradient Brownian flows on spheres.
For more recent work see [63a], [63b].
§2. Mean exponents
A. The mean exponents XL are considerably easier to study than the actual
exponents themselves. Since they are given by (128):
).. L(x) = lim 1/t log det T xF 1( -,w) t-+OO
their existence depends only on the usual additive ergodic theorem, rather than
the more sophisticated multiplicative theorem
The covariant equation (77): Dvt = VY(vt) o dzt can be interpreted after
parallel translation back to x0 and the addition of an S.D.E. for the horizontal
lift of Xt as an equation on OM x IL(IRn;IRn). To describe det TFt we can
therefore use the Ito formula, and with the notation used in §SA of Chapter II
we must calculate d/dt log det &S(t,-)at t = 0 and its second derivative. In fact
172336
it is a classical result for flows of ordinary differential equations, (the
continuity equation), which is left as an exercise, that
d/dt det oS(t,-) = div Ye(S(t,-)e).det oS(t,-).
where Ye is the vector field Y(-)e. See Lemma 2B of Chapter V.
Thus, at t = o,
d/dt log det oS(t,-) = di v Ye
and
d2/ctt2log det oS(t,-) = <V(div Ye),Ye>·
Consequently
log det T xFt = Ito L div xP(xs)dBPs +Ito div A(xs)ds
+ ~ Ito L d(V div XP(xs),XP(xs)>ds
(133)
(134)
Since M is compact L: div XP(s) is bounded, so the Ito integral in (134) is a
time changed Brownian motion B"t(t)• say with "t(t) :s; const.t for all t.
Therefore t-1B"t(t) -+ 0 as t -+ oo. Applying the ergodic theorem, with the
notation of §1A, for p almost all x:
lim 1/t log det T X Ft = J (div A(x) + ~ L <V div xP(x),XP(x))) p(dx)IP(dw) MxQ
i.e.
:>..r = JM div A(x) p(dx) +~LIM <V div XP(x),XP(x))p(dx) (135)
This is a special case of formulae by Baxendale for sums of the first k
exponents in [12 ].
For a Brownian flow, i.e. when A = ~ 6, the invariant measure pis just the
normalized Riemannian measure. We can use the divergence theorem, equation
(1.5), to dispose of the first term of (135). For the second term we can
integrate by parts: in general if f:M -+ !R is c1 and Z is a c1 vector field
applying the divergence thoerem to fZ together with the formula
div fZ = f div Z + (Vf,Z)
yields
JM f div Z dx =- JM (Vf, Z)dx
Thus
(136)
(137)
173337
A';= - (2!MI)-1 "J M '; (div XP(x))2dx (138)
where IMI denotes the volume of M.
B. From (138) we see that in the Brownian case A'; s 0 with equality if and
only if div XP = 0 for each p. More general results are obtained by Baxendale in
[12a] and we consider a simple version of those, assuming now that A is non
degenerate.
For Borel probability measure A,Jl on a Polish space X define the relative
entropy h(A;Jl) e IR (;:: O) u + oo by h(A;Jl) = oo unless Jl s :>,and
f X dJl/dA I log dJl/dA I d),< oo (140)
in which case
h(A;Jl) := f X (dJl/dA log dJl/dA)d:>,
= f X (log dJ.l/dA) dJl (141)
To see that h(A;Jl);:: 0 observe that x-+ x log xis convex on (O;oo) so by Jensen's
inequality if h(A,Jl) < oo then
h(:>,;Jl);:: (f dJl/d), d).) log (f dJl/d), d).)= 0
with equality if and only if Jl = :>,.
We shall be particularly interested in the case where :>, = p, the invariant
measure of our S.O.S. on M, and Jl = Pt where Pt is the random measure on M
defined by
Pt(w)(A) = p(Ft(-,w)-1(A))
for A a Borel set in M and we Q.
Following Baxendale [12a] and LeJan [69] we will consider h(p;pt):
Theorem 28 For a non-degenerate system
A'; = - 1/t IEh(p;pt) (142)
Consequently:>,'; s 0 with equality if and only if p is invariant under the
sample flow t-+ Ft(-,w): t;:: 0 for almost all tJ) t Q.
Proof
Let ), denote the Riemannian measure of M, and for t ;:: 0 abuse notation
so that Pt(dx) is written Pt(X)dx: by standard results, since A is elliptic,
Pt(-,w) : M-+ IR is smooth and positive. Then p 1(x) = p 0 (y)(det T yFt)-1 for
y = Ft-l(x), and so
IEh(p;pt) =IE f (log Pt(X)- log p 0 (x))pt(X)dx
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= lE J (log Pt(F t(x)) - log p 0 (F tCx))) p 0 (x)dx
= lE J (log p 0 (x)- log det T xFt - log p 0 (Ft(X)))p 0 (x)dx
=- lE J (log det T xFt)p 0 (x)dx
= -tAL: by (134) and (135). I I
In [12a], Baxendale gives a formula for h(p;p 1) analogous to (134) and
using analogous computations which can be based on the continuity equation
d/dt p 0 (S(t,x)e)det oS(t,-)) = div(p 0 Y(-)e)(S(t,x)e).det o S(t,-) (143)
See equation (285) in §2B of Chapter V below.
As he points out some non-degeneracy conditions are needed: in the
completely degenerate case of an ordinary dynamical system, when p is the
point mass at a source A:L; > 0, and when it is a point mass at a sink A:L; < 0,
while in either case h(p;Pt) = 0.
c. For a gradient Brownian system (see §7 Chapter II) it is possible to get a
neat upper bound. For <p : M-+ IRm the isometric embedding, so that XP = VcpP,
equation (138) gives, by the Cauchy-Schwarz inequality,
A L: = - 1/(2!MI) J M L:p (6<pP)2dx
:S- 1/(2!MI) L: (J M cpPt:.cpP dx)2 (J M (cpP)2 dx)-1.
By a translation in IRm we can assume that J cpP dx = 0 for each p, i.e. that
cpP is orthogonal in L2 to the solutions toM= 0 (i.e. the constants, since M is
compact). Therefore, with an integration by parts
A:L; :S 1/(21MI) L: J M !XP(x)l2x dx J M cpP t:.cpP dx/ J M(<pP)2dx
:S 1/(2!MI) x (leading eigenvalue of 6) f M L:p IXP(x)l2 dx.
Since L:p IXP(x)J2 = n for all x, this yields Chappell's result [25]:
Proposition 2C Let J.l be largest non-zero eigenvalue of Li. Then
1/n A:L; :S ht. (144)
Moreover there is equality if and only if~ 6<pP = J.l<pP for each p, [25],
[22].
Such embeddings have been studied by Takahashi, see [65] Note 14. The
simplest examples are the spheres sn with their standard embeddings in 1Rn+1.
Another example is the torus in IR4 which is the image of
cp: s1 (1/2) x s1(1/2)-+ IR4
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~(u,v) = (1/2 cos u, 1/2 sin u, 1/2 cos v, 1/2 sin v) (145)
Here A.:L = -2.
Note that by (74), in the gradient Brownian case, (138) is just
A.:L = - 1/(21MI) I M I trace o::xl2 dx (146)
For the sphere sn(r) of radius r in 1Rn+1 this shows directly that
1/n A. :L = - ~ n/r2
since
o::x(v,v) = - ..!_ lvF (x/lx!) r
(147)
(148)
§3. Exponents for gradient Brownian flows: the difficulties of estimating
exponents in general
A. For a gradient Brownian flow, if Vt = T x F1(v 0 ), equation (95) reduces, 0
when lv0 1 = 1, to
dlvtl2 = 2(Vt,VX(Vt)dBt>- Ric(vt,Vt)dt + :Lp(VXP(vt),VXP(vt))dt (149)
giving
1/t log lv11 = 1/t It0
<1ls.VX(fts)dBs>- 1/(2t) It0
Ric(fts,fl.s)ds
- 1/t It0
:Lp <1l.s• vxP(1\s)>2 ds
+ 1/(2t) I 10
:Lp (VXP(vs),VXP(vs))ds (150)
where 1\s = Vs/lvsl in the sphere bundle SM for SxM = (v e T xM: lvlx = 1). By
(70) for v e T xM
vxP(v) = Ax(v,ep - xP(x))
so
and
Thus
176340
t
lim .!..loglvtl =lim.!.. J ~I cxx<Tls,-)12- lcxx<Tls•Tls)l2- ~ Ric (Tls•Tls)ds (151) t t
0
(almost surely).
This can be modified by the use of Gauss's theorem that for vET xM
Ric (v,v) = -1 cxx(v,-)12 + <cxx(v,v), trace cxx> (152)
so as to get an expression entirely in terms of the second fundamental form
and process Tlt: 0 ~ t <co.
so
For sn(r) in IRn+1, if u, vET xsn(r)
(n-1) 1 x Ric (u,V) = - 2- (U,V) and CXx(U,V) = - r <U,V>r
r
lim 1/t log lvtl =- ~ n/r2.
t~co
Thus
A 1 = - ~ n/r2 ( 153)
and so A1 = 1/n AL by (147). This shows that all the exponents for the spheres
are the same. Bougerol [19] has shown that among all hypersurfaces it is only
the spheres which possess th1s property.
B. The process Tlt: 0 ~ t < co 1s g1ven by an S.O.S. on SM. In fact this is
just d'Tlt = P(Tlt)&X(Tlt) o dBt where P(71) is the orthogonal projection in T xM of
T xM onto TSxM· By compactness of SM it will possess ergodic probability
measures which project onto p. If vis one of these we get for v-almost all v0
lim 1/t log lvtl = JSM nlcxx(Tl,-)12- lcxx(Tl,Tl)l2- ~Ric (Tl,Tl)v(d'Tl) (154)
As the right hand side of (154) varies over the ergodic measures v which
project onto p it gives a subset of the set lyapunov exponents, sometimes
called the Markovian, or deterministic, spectrum. They correspond to
elements of the flltration which are non-random: see [23], [62] for details. In
particular the top exponent Allies in this set.
Thus if the integrand of (154) is strictly negative for all 11 E SM the top
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exponent will be negative. Using (152) it is straightforward to see that this
holds for hypersurf aces if the pri nci pa 1 curvatUt~es 2 1 (x), ... ,.2n(x) at each point
x satisfy
x e M, j = 1, .•. ,n
for some e > 0, see [26]. This is a convexity condition. It seems reasonable
to guess that X 1 < 0 when M is the boundary of a convex domain.
B. Formula (150) is a version of Carverhill's VE~rsion of Khasminski's formula
(see his article in [B]). From it we get more general versions of (151) and
(154). A major difficulty in extracting information from (154) is the lack of
knowledge of the behaviour of the invariant measures v, in particular lack of
knowledge about their supports (one cannot expe~ct the infinitesimal generator
of 'Tlt : t 2:: 0 to be elliptic or even hypoellipt·ic in general). Control theory
gets involved here: see the article by Arnold et al in [8], and also more recent
work by L. Arnold and San Martin.
C. Rather than considering the process 11t: t 2:: 0) on SM it is often more
convenient to take its projection onto the projective bundle PM which is simply
the quotient of SM obtained by identifying antipodal points in each fibre SxM· It
is shown in [12a] that given ellipticity of A (for example) there is an invariant
measure v for this process such that with vt its shift by the flow of the
process on PM
IEh(v;v1)- h(p;p1)) ~ n X1- XL (155)
where his the relative entropy as in §28. Using this Baxendale showed that all
the exponents are equal given some non-degeneracy of A (e.g. ellipticity), if
and only if there is a Riemannian metric such that the sample flows Ft(-,w)
are conformal diffeomorphisms. See also [19].
D. For gradient Brownian flows the exponents and their multiplicities are
geometric invariants of the embedding of M into IRm. We have seen that in
general there are non trivial filtrations of tangent spaces T xM. These are
dependent on the embedding and the particular sample path: it is rather
difficult to imagine what, necessarily long time, property of the sample path
will determine the position of say v(2)(x,w) in T xM·
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§4. Exponents for canon1cal flows
A. Consider the canonical flow on the orthonormal frame bundle OM of the
Riemannian manifold M, (or on SOM if M is orientable, for some orientation).
There is a natural metric on OM defined by requiring that the trivialization tp
of TOM, given by equation (107) in §9 Chapter II gives isometries 'Pu: T uOM -t
IRn x .Q.(n) for each u e OM. Here the inner product on .Q.(n) is taken to be
(A,B) = - ~ trace AB (156)
for A,B e .Q.(n) identified as skew-symmetric matrices. The factor of ~ has
some advantages e.g. if e e IRn
IAel ~ IAIIel (157)
with this definition. (A disadvantage is that it was not used in [24]). The
corresponding measure on OM is sometimes called the Liouville measure.
Since T uTI : T uOM -t T n(u)M is an isometry on the horizontal subspace HuOM and
vanishes on its orthogonal complement, we see TTT maps the Liouville measure
onto the Riemannian measure of M. Also by the invariance ()oTRg = ad(g-1)offi
of connection forms and the invariance under ad(g-1) of the given inner
product on .Q.(n) it follows that the Liouville measure is invariant under the
right action of O(n) on OM.
It is a standard result, observed by Malliavin, that the canonical flow has
sample flows Ft(-,w) which preserve the Liouville measure. Rather than check
that div XP = 0 for each p we can see this from the Stratonovich equations
(121a) and (121b)
ds t = At o dBt
dAt = Ut -1 R(Ut o dBt, Ut S t)Ut
of §9C Chapter II for s t = e( oF t(V)), At = w (oF 1(V)). Indeed the equation for s t
involves only At and conversely, so the trace of the right hand side considered
as a linear transformation of <st,At) vanishes identically. Therefore the
Stratonovich equation for det oFt(-) shows that the determinant is identically
1, and so the Liouville measure is preserved.
Our Lyapunov spectrum wm be taken wHh this as basic measure. However
in general it will not be ergodic: for example it will not be if M is the product
M1 x M2 of two Riemannian manifolds, or when M is flat (i.e. has vanishing
curvature). In the latter case we noted in §98 of Chapter II that OM is foliated
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by horizontal submanifolds: each of these will be invariant under the flow.
More generally the holonomy bundle, see [64], is invariant.
C. Since the system can degenerate we must first show that the exponents
:>..(r)(u) < ••• < :>..( 1)(u) can be taken to be independent of u e OM. Here, and for
the rest of this discussion of canonical flows we are following [24]. To do
this observe (as in the proof of Theorem 4C, Chapter II) that forge O(n)
Ft(U,W)·g = Ft(U·g, g-1w) (158)
and so for V e TOM
TRg(TFt(V,w)) = TFt(TRg(V), g-1w) (159)
The measure one subset r of OM x Q in Theorem 18, consisting of points for
which convergence to the exponents occurs can therefore be taken to be
invariant under (u,w)-+ (u·g, g-1w) for g e O(n), with corresponding invariance
for the filtrations i.e. vm(u,w) = y(j)(u·g, g-l.w), and so for the exponents:
A.(j)(u·g) = A.(j)(u), since they are non-random. Thus we obtain maps A.(j)0 : M-+
IR with A.(j) 0 (n(u)) = A.(j)(u) for u in OM, defined almost surely. These are
measurable. Also since each A.(j) is invariant under ~t:OM x Q-+ OM x Q, we
have
:>..(j)(u) =IE :>..(j)(Ft(u,w)) =IE :>..(j) 0 (n F1(u,w))
=Pt:A.(j)o(X)
for x = n(u), where (Pt: t 2: 0) is the heat semigroup forM (solving o/ot = it.).
Thus Pt A.(j)0 is independent oft, and so A.(j)0 is constant (for example by the
ergodicity of the Riemannian measure: but this itself is usually proved by
observing that Ptf independent of t implies 6P1f = 0 for t > 0, since Ptf is c2
for t > 0, which implies Ptf is constant for each positive t , which implies by
strong continuity of Pt in t that f is constant).
D. From our equations (122a,b) forst= e(&Ft(V 0 )) e IRn and At= ro(&Ft(V0 )) e
o(n.) we could write down an expression for logl&Ft(V 0 )1 = i log(lst12 + 1Atl2).
However that does not seem very illuminating, and we shall resist doing so
(but see equation (172) below when dim M = 2). To start with we shall just
consider the horizontal component St· For this set Vt = Tn(&Ft(V 0 )), so Vt =
ut<st) and Vt e T xtM for Xt = n(u 1) the Brownian motion induced on M. In
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particular lvtlxt = l~tl· By (122a)
1~ 1 12 = 1~0 12 + 2 It 0 <~s• AsdBs>- I\ Ric(vs,Vs)ds + 2 I\ lAsF ds (160)
log l~tl =log 1~ 0 1 + It0 <~s/l~sl, As/l~sl dBs>- ~ It0
Ric(vs/lvsl, Vs/lvsl)ds
+It 1Asi2/J~sl2 ds- It 1As~sl2/l~sl4 ds (161) 0 0
(at least until the first hitting time "t of 0 by ~t).
Therefore by (157) and the observation that !Ael =!AI lei when n = 2
log l~tl2: log 1~ 0 1 + Mt- ~ It Ric (vs/lvsl, Vs/lvsl)ds (162) 0
with equality when n = 2, where Mt: t 2: 0 is the local martingale
t Mt = I
0 <~ sll~ sl, As/1~ sl dBs> ( 163)
Now Mt: T 2: 0 is a time changed Brownian motion and for ~t to vanish in
finite time "t (assuming ~ 0 * 0), we would have to have
limt-n- Mt = -oo. Then limt-+"t- Mt = oo and so
limt-+"t- log(loFt(V0 )1) 2: limt-+"t- log l~tl = oo
which cannot be true for finite "t. Thus l~tl never vanishes and (161) holds for
all time.
Theorem 40 [24]. Let Ric(x) = sup Ric(v,v) : v e T xM and lvl = 1 for each
x e M. Then the top exponent A 1 of the canonical flow satisfies
A 1 2: lim t-+oo 1/t log lvtl2:- 1/(21MI) I M Ric(x)dx (164)
Proof: Since Mt is a time changed 1-dimensional Brownian motion
limt-+oo 1/t Mt ~ 0 ~ limt-+oo 1/t Mt
Therefore by (162)
limt-+oo 1/t log lvtl2: lim(- 1/(2t) J1 Ric(vs/lvsl, Vs/lvsD) ds)
0
2: - 1i m 1/(2t) f t Ric(xs)ds 0
= - 1/(21MI) I M Rlc(x)dx
almost surely, by the ergodic theorem, since (1/IMI x the Riemannian measure)
is ergodic for Brownian motion. II
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Remark 40 For dim M = 2 the Ricci curvature is essentially the Gaussian
curvature Kp(x) for each x. The Gauss-Bonnet theorem states that
1/(211) I M Kp(X)dx = x(M) (165)
where X(M) is the Euler characteristic of M, a topological invariant (e.g. X(S2)
= 2, xcs1 X s1) = 0). It is proved in Chapter VI below. From this, (162), and
the argument above: if dim M = 2 then for p 0 G' almost all (x,w)
A 1 2:: iiillt-+oo 1/t log lvtl = limt-+oo 1/t Mt - ( 11/!MI) X(M)
;::: - (11/IMI) X(M) (166)
Also
limt-+oo 1/t log lvtl =lim 1/t Mt - (11/IMI) X(M) :S -(11/IMI) X(M) (167)
(The~ in the corresponding formula in [24 J should not be there).
From (167) we get
A1 =lim 1/(2t) log(l~tj2) + 1Atl2 =lim 1/t (logl~tl + ~ log(1 + 1Atl2 /l~tl2 ))
:S -(11/IMI)X(M) + iilli 1/(2t) log(1 + !Atl2/lstl2 ).
Since A 1;::: 0 this shows: for dim M = 2
lim 1/t log (1 + 1Atl2/lstl2);::: (211/IMI) X(M). (168)
E. Next we consider the case dim M = 2 in more detail.
Write k(x) for the Gauss curvature Kp(x) (with P = T xM) so that
Ric ( u,u) = lul2 k(x) (169a)
and
R(u,v)w = k(x)((w,v)u- (w,u)v (169b)
for u,v,w in T xM· The following formulae are given for completeness. They
come from (160) and (122b): the rather straightforward proof is left an
exercise; there are details in [24] (using the scalar curvature S(x) = 2k(x))
Is tl2 = lsol2 + 2 It 0 <s s•AsdBs> - f t
0 k(xs) Is sl2 ds + 2 f \ 1Asl2 ds ( 170a)
IAtF = IA 0 12- 2 J1 k(xs)<ss.AsdBs>- 2ft k(xs) 1Asl2 ds
0 0
+ J1 dk(usAsss)ds + J1
k(xs)21ssl2ds (170b) 0 0
The following formulae from [24] are useful:
Proposft1on 4E
For dim M = 2 there is the Stratonovich equation
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diAtl2 + k(Xt) o dis tl2 = 0 ( 171)
and the Ito equation
IA1l2 + k(x1)1s 112 = IA 0 12 + k(x 0 )1s 0 12 + f t Is sl2 dk(usdBs)
0
+ ~ J1 lssl2 ~k(xs)ds- J1
dk(usAsss)ds (172) 0 0
Proof The Stratonovich equations (121a,b):
give
dst =At 0 dBt and dAt = Ut-1 R(ut o dBt, Utst)Ut
dis 112 = 2<s t· At o dB1>
d1Atl2 =-trace At Ut- 1 R(Ut o dB, Uts1)ut
!. (Ut -lR(Ut o dBt.Uts t)Utep,A1ep) p=l
(173)
!. k(Xt) <uteput,st><utodBt,utAtep> - <ut ep, ut o dBt><ut s t•Ut At ep> p=l
(174)
Equation (171) follows immediately. On integrating it by parts and then using
the Ito formula for k(xt) (and hoping the use of d for stochastic differentials
as in d(k(xt)) and for ordinary differentials as in dk will not cause confusion):
IA112 + k(xt)lstF- IA 0 12 + k(x 0 )ls 0 F
= Jt d1Asl2 + k(xs) o dlssF + lssl2 o d(k(xs))
0
= ft lssl2 0 d(k(xs)) 0
= J t Is sl2 dk(us o dBs) + ~ "f t Is sl2 ~k(xs)ds + ~ f t dis sl2dk(usdBs) 0 0 0
giving (172) by (173). II
Theorem 4E
When dim M = 2 and k(x) > 0 for all x
)...1 s 1/(41MI) fMlVk(x)l/.fk(x) + l~k(x)l/k(x)dx (175)
f.r..Q.Qf
Since k(x) > 0 we can take .fIAI2 + k(x) Is 12 as the norm of (s,A) when
computing the exponents. Write it as ll(s,A)I!. By (172)
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log ll<st.At)ll =log ll<s0 ,A0 )11 + ~ Jt lssl21 ll(ss.As)ll2 dk(usdBs) 0
+ 114 f t 0
Is sl2 I ll(s s•As)ll2 t.k(xs)ds - ~ f \ dk(usAss s)dsiiKs s•As)ll2ds
- 114 J t Is sl4 l(dk)x 12 lll(s s•As)ll4ds ( 176) 0 s
Since
ldk(usAss s)l = (Vk(xs),usAss s> :s 1Vk(xs)IIA5 lls sl
:s ~ 1Vk(xs)l.fk(xs)-111(s s•As)ll2 (177)
and the coefficient of the Ito integral in (176) is bounded, the ergodic theorem
gives the result. I I
This corrects the upper bounds in [24 ]. It must be possible to do better.
F. Next we consider the case of constant curvature. An important point here,
and later, is the idea of a covering p: M-+ M. This is a C00 map of manifolds
which is surjective and such that each x e M has a connected open
neighbourhood U with p mapping each component of p-1cu) diffeomorphically
onto u. The typical examples are p: s1-+ s1 given by p(eie) = e2i9 and
p:IR -+ s1 given by p(e) = ei e. The covering is Riemannian if M and M are
Riemannian and T zP: T 2 M-+ T p(z)M preserves the inner product. Clearly if M
is Riemannian and p is a covering map then we can define a Riemannian metric
on M so that p becomes Riemannian. In general coverings have the path
lifting property; if d : [a,b)-+ M is a continuous path and z e p-1(d(a)) then
there is a unique continuous d- : [a,b) -+ M with d-(a) • z and p o d- = d. The
lifting gives a continuous map from the space or continuous paths in M starting
from <5(a) to the corresponding space of paths from z in M. Thus stochastic
processes can be lifted from M toM. Also it is easy to construct an S.O.S. -(Y ,z) p-related to a given one (Y,z) on M.
For a Riemannian covering a Brownian motion on M maps by p to a
Brownian motion on M: to see this choose an S.O.S. (Y,z) on M which has -Brownian motions as its solutions. The lift (Y ,z) will then have Brownian
motions on M as its solutions since the condtions of §4A Chapter II for this to -happen are purely local, and locally (Y ,z) and (Y ,z) are the same, as are M and -M. Since (Y ,z) and (Y,z) are p-related the result follows by §3D Chapter 1.
(For a generalization of this see [43], p. 256.) Since curvature is a local
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property p will also map the curvature tensors of M to that of M.
Given the equation dxt = X(xt) o dBt + A(xt)dt on a compact Riemannian M
and a Riemannian covering p : M -+ M, it is immediate that the Lyapunov
filtrations and any stable manifolds lift to corresponding objects for the lift of
the SDS: the filtration ofT 2 M will map by T zP to the filtration forT p(z)M and
will exist when the latter exists, with the same exponents, and p will map
stable manifolds to stable manifolds as a covering. In particular Nl. need not be
compact.
The other two main ingredients we need are:
(i) there exists a covering p : Nl. -+ M with Nl. simply connected, and this is
essentially unique
(ii) if M is simply connected and of constant curvature k then: M is isometric
to IRn if k = 0, to sn if k = 1, and to hyperbolic space Hn if k = 1. (e.g. see [64 ]).
If we now note that for a Riemannian covering p: M-+ M the map u-+ Tpou
:OM-+ OM is also a covering, we see that to investigate the Lyapunov exponents
and stable manifolds when M has constant curvature k = +1 or k = -1 it suffices
to take M = sn or M = Hn.
G. Suppose now that M has constant curvature k. Equations (173) (174) are
valid with k(xp) replaced by k, because of formula (9) for the curvature. Thus,
[24 ],
d!Atl2 + k dl~tl2 = 0
whence
(178)
When k > 0 we see immediately that :>-.1 = 0, whence :>-.1 = A.L:, and so all
the exponents vanish given constant positive curvature.
Fork< 0 we see that except perhaps for some exceptional V 0
:>-. 1 =lim 1/t log lstl =lim 1/t log IAtl
t-+oo t-+oo
and in particular these limits exist.
From ( 164) this yields
lim 1/t log lstl ~- (n-1)k/2 > o t-+OO
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so that almost surely
~ 1Atl211stl2 = -1< + ~ 0A0 1 + 1<ls0 12)11stl2 = -1<
From (161) (162) we see 1mmed1ately that if .Q.Lm M = 2 then for k < o (179)
G. For constant negat1ve curvature since ).1 > 0 and A:E = 0 there must be
some negative exponent and correspondingly sCtme stable manifolds. We will
investigate these for k = -1 and dim M = 2 following [24]. By the discussion in
the previous paragraph we need only take M to be the hyperbolic plane H2, even
though it is not compact.
In §SC of Chapter II we described H2 as the hyperboloid (t,x,y) e IR3: t ~ 1
and x2 + y2- t2 = -1 with Riemannian metric induced from the Lorentz metric
of IR3.
Writing N = (1,0,0) the tangent space TNH2 can be identified with
0) x IR2 in IR3. For v = (v1, v2) in IR2 the path 'tv: IR-+ H2 given by
"tv(<X) = (cosh(lvi<X), v1/lvl sinh(lvi<X), v2/lvl sinh(lvi<X))
has
d~ lv(<X) = (lvl sinh(lvi<X), v1cosh(lvi<X), v2 cosh(lv!<X))
so that
d 1-d "tv(<X)I ( ) = lvl.
a lv <X
Differentiating this we see D/o<X d/d<X "tv(<X) is orthogonal to d/d<X "tv(<X) and
so vanishes by symmetry. Thus "tv is a geodesic through N and using our
identification of TNH2 wHh IR2
expNv = l v< 1).
For V e T viR2 ::::: IR2 • with v * 0
DexpN(v)(V) = (<v,V)/Ivl sinhlvl, (sinhlvl)DP(v)(V) + (<v,V)/!vl)(coshlvl)(v/lvl))
E IR X IR2
where P: IR2- 0)-+ s1 is P(v) = v/lvl. Thus
! D expN(v)(V)J2 = (V ,P(v))2 + IDP(v)(V)j2(sinhlvl)2
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since DP(v)V is orthogonal to v in IR2. This gives the induced metric in the
chart given by expN i.e. in normal co-ordinates at N
(V,W>v = (V,P(v))(W,P(v)) + sinh(lvi)2(DP(v)V,DP(v)W)IR 2 (180)
which is most easily considered in polar co-ordinates: in classical notation
ds2 = dr2 + (sinh r)2de2 • (181)
(One way to see this is to interpret (18)) as meaning that if a curve 6 in
M is given in normal co-ordinates by 6(t) = Crt cos et , rt sin et) then
lo(t)G(t) = r; +(sinh rt)2 ·a;).) Let oo be the open unit disc in IR2. Define f: D-+ IR2
by
f(r,e) = (2 tanh-1r,e)
in polar co-ordinates. The metric (181) on IR2 induces the metric
ds2 = 4(1-r2)-2 (dr2 + r2de2) (182)
on oo, or in Cartesian co-ordinates
ds2 = 4(1-r2)-2 (dx2 + dy2). (183)
The disc with this metric is the Poincare disc model of H2. It represents N as
(0,0), but since the subgroup G of the Lorentz group acted transitively on the
hyperboloid as isometries we can compose expN of: D0 -+ H2 with an isometry
to get an isometry which maps (0,0) to any given point of H2.
There is also the representation of H2 by the upper ~-plane
u = (x,y): y > 0 in IR2. For this choose some point c of s1. Then there is an
analytic diffeomorphism de: U-+ oo
dc(z) = c(z-i)/(Z+i) (184)
which maps the closure u to the closed disc D with the point at infinity in U
mapped to c. The metric induced on U is
ds2 = y-2(dx2 + dy2) (185)
The disc model shows how to talk about "points at infinity" on H2 :they can
be taken to be the points of s1. For c e s1 write uc for U when de has been
used to give it its metric.
H. Since any p e H2 can be identified with (0,0) in D we can identify the points
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of 'the circle at oo' , N00 , in H2 with ~ rays 1 emanating from p and
parametrized by arc length. The Buseman function of such 1 corresponding to
cis
J3p (c,-): H2-+ IR
for
J3p(C,Z) =lim (t - d(Z,"t(t))) 1-+00
(186)
(Since t -+ t - d(z,"t(t)) is increasing and bounded above by d(z,p) this limit
exists). This is sometimes given the opposite sign.
In the model uc with p = (0,1) we have "t(t) = (o,et) and if z =
(x,y) e uc then
t- logy :S d(z,"t(t)) :S t-log y + e-2t lxl
since logy is the distance of z from the line (cx,1): ex e IR. Thus in this case
J3p(c,z) =logy (187)
Lemma 4H [24]
Let zt : t 2:: 0 be a Brownian motion starting from p. Then with
probability 1
and
(i) z00 (w) =lim Zt(w) e N00 exists t-+OO
(ii) lim 1/t J3p (z 00 (w), Zt(w)) = ~ t-+OO
(188)
Proof Part (i) is a very special case of Prat's result for not necessarily
constant curvature. In our case it follows because in oo our Brownian motion
is just the time change of an ordinary Brownian motion in IR2, and the latter
almost surely leaves 0° in finite time.
For (ii) it is enough to show that
IP1/t J3p(C,Zt(w))-+ ~ I z00 (w) =c)= 1.
To condition z to tend to c we can use the Doob h-transform. Now, as
described in [81] 2X9 the standard Brownian motion IR2 conditioned to exit
from oo at a point c of s1 is the h-transform of that Brownian motion, h
transformed by the Poisson kernel
h(z) = (1- lzF)/Ic-zl2 lzl < 1.
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This means it has the law of the diffusion process with generator ~t:. + 'Vlog h
for t:., 'V the Euclidean operators. Since time changing commutes with our
conditioning the hyperbolic Brownian motion of M conditioned to tend to c as
t-. oo is a diffusion process with generator ~t:. + 'V log h where now t:. and 'V
refer to the hyperbolic metric.
In the model Uc the Laplacian is given, for x1 = x, x2 = y, by
M(z) = 'f g(z)-112 a;axi g(z)~ gii(z) of /oxi J=l
= y2 (o2f ;ax2 + a2f ;ay2)
while his represented by h ..... for h ..... = h ..... • de i.e.
h ..... (z) = 1/4(1z+il)2- lz-il2) = y for z = x + iy.
Thus
V log h ..... (z) = (0, y2. 1/y) = (O,y)
(189)
and the conditioned diffusion can be represented by Zt = (Xt•Yt) for
dx t = Yt dB 1t, d!Jt = Yt dB2t + Yt dt (190)
where (B 1t, s2t): t 2:: 0 is a Brownian motion on !R2. Then Yt =Yo exp(B2t +
~ t) and so (ii) follows by (187). I I
We can now give the basic result from [24] on the stable manifolds of the
canonical flow on OM for hyperbolic space:
Theorem 4H [24] For M = H2 take u e OM. Let Ft(-,w):OM -+ OM be the
canonical flow. Then for almost all w e Q the following holds:
The limit c(w) =lim TT F1(u,w) exists in N00 and if 'Y'(u,w) is the 1-+00
submanifold of OM given by Tg o u s.t. g: uc -. uc is a horizontal
translation) then for u' e V(u,w)
lim 1/t log d(Ft(u,w), Ft(u',w)) = -~ t-+OO
and for all other frames u'
lim. 1/t log d(Ft(u,w), Ft(u',w)) 2:: 0
t-.oo
(191)
(192)
Proof: Choose w e Q so that the conclusions of Lemma 4H are true, and so that
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the flow Ft(-,w) exists and satisfies Ft((Tg) o u,w) = Tg o Ft(u,w) for all
isometries g of M. The latter is possible either by general principles, because
the canonical S.O.S. is invariant under the action of such Tg, or by noting the
special properties of the flow Ft(-,w) on OM when OM, or rather SO(M), is
identified with our subgroup G of the Lorentz group: see Remark 4H(i) below.
Then c = z00 (w) exists. We will work in uc.
It is necessary only to consider oriented frames i.e. restrict ourselves to
the component SOM of OM. Such a frame at (x,y) e uc can be identified with a
tangent vector to uc of unit Euclidean length. Using this we shall write frames
as (x,y,A) e uc x s1. Let d"' be the metric on SOM which is the product of the
Euclidean metric on uc with the standard one for s1. Over the compact subset
W of uc
W = (x,y) e uc: lxl + 11-yl ::s ~ this will be equivalent to the standard metric of OM described previously (or to
any other metric).
Set (Xt,Yt,At) = Ft(u,w).
If u' e 'V'(u,w) there exists a e IR with
F1(u',w) = (Xt +a, Yt•At) t ~ 0
Horizontal translation in uc is an isometry and so is the dilation (x,y)-+ (exx,
exy) for ex> 0. Therefore
d(Ft(u,w), Ft(u',w)) = d((O,yt,At), (a,yt,At))
= d((0,1,At), (a/yt, 1, At)) (193)
since isometries on M induce isometries on OM. For sufficiently large t both
(0,1) and (ayt-1,1) lie in W, and sod may be replaced by d in estimating (193)
for such t. However .....
d ((0,1,A1), (a/yt, 1, At))= lal/lytl
and by Lemma 4H(ii) and equation (187)
1 i m 1/t log (lal/lytl) = - ~ t-+oo
This proves ( 191).
(194)
For (192) first suppose u' = (exx0 , exy0 , A0 ) for some ex> o, ex* 1 where u =
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(x 0 ,y0 ,:>.. 0 ). Then F t(u' ,w) = ( «XXt, o:yt,:At), giving (by a hori zonta 1 translation)
d(F1(u,w), F1(u',w)) ~ d((O,o:yt), (O,y1))
= d((O,o:), (0,1))
from which (192) follows.
Combining this with (191) we see the same holds for any u' which is
obtained from u by the action of the isometries of M generated by the
horizontal translations and the dilations (x,y) ... (o:x, o:y) of He. These
isometries correspond to a subgroup Gc, say, of G when we use the
identification of SOM with our subgroup G of the Lorentz group. The group G
itself was identified with isometries of Min the hyperboloid model and in this
model it is easy to see that Gc is precisely the subgroup of G which leaves the
point at infinity c fixed (the latter subgroup is just the natural embedding in G
of the identity component of the Lorentz group of the 1 + 1-dimensional space
time acting in the plane orthogonal to c in IR3: this is two dimensional as is Gc
and the former is known to be connected).
For other u' in SOM = G there is the isometry corresponding to
g = u·u-1 which sends u to u'. Since Ft(u',w) = g Ft(u,w) and g is not in Gc
lim Ft(u',w) *c. 1-.00
Consequently
d(Ft(u,w), Ft(u',w)) ~ d(TT Ft(U,W), TTFt(u',w))
~ d(TT Ft(u,w), (x,1):x E IR))
or sufficiently large t. This is just J3p( c, TT ,F 1(u,w )) for p = (0, 1) by ( 187). Thus
in this case, by ( 188)
lim 1/t log d(Ft(u,w), Ft(u',w)) ~ ~. II
This theorem, together with the fact that we know there must be at least
one negative exponent with corresponding stable manifolds, shows that there
is precisely one, namely-Land that the stable manifold through u is Y(u,w).
Consequently the multiplicity of the exponent - ~ is dim'V'(u,w) i.e. 1. Since dim
OM= 3, and :>..:L = o and :>..1 =~by (179), because 2(~) + (- ~) = ~ * o there must
be another exponent. It can only have multiplicity 1 and it must be 0. Thus
the exponents for the canonical flow on H2 are-~. 0, ~·
As for the filtration of TuM we know that v(3)(u,w) = T uY(u,w). It will
191355
now be no surprise that y(2)(u,w) is the tangent to the orbit of Gc (i.e. the
tangent to the coset Gcu in G): for a proof see [24]. More detailed information
'stability' properties of the flow can be found in [24] and [10], especially the
latter.
Remark 4H(j) Identifying SOM with G the canonical S.D.E. becomes a left
invariant stochastic differential equation
dUt = X( Ut) o dBt
with X(1)(0 = .t for s in IR2 as in equation (56) of Chapter II §SB. The flow is
then Ft(u.w) = u.g1(w) where gt : t 2: 0 is the solution starting from 1.
Equation (178) showing that IAtF- lst12 is constant follows from the invariance
of the Cartan-Killing form: see [64] p. 155. The metric we have taken on SOM
corresponds to a left invariant metric on G so our exponents are measuring
how right multiplication by gt(w) spreads out or contracts the space (at least
infinitesimally). Use of the Lie group structure of G gives a good way to obtain
the result about the exponents given above and especially for their higher
dimensional analogues. This is carried out in [10 ]. See [71 ] , [73] for
earlier work for symmetric spaces.
The vanishing of the exponents for sn comes out particularly simply by the
corresponding representation of sosn as SO(n+1). This time the metric on
SO(n+1) is bi-invariant and so F1(-,w) consists of isometries: this is the reason
for the constancy of lst12 + 1Atl2 in equation (178) fork= 1.
Remark 4HOO The projections onto M of the stable manifolds 'V'(u,w) are
horocycles. In the disc model the horocycle Hp(c) for p e M and c on the circle
at infinity is the circle tangent to s1 at c which goes through p. The
horocycles are precisely the level surfaces of the Buseman functions defined
by (186). Equivalently they can be defined as the boundary of the horoballs
defined as the union U Bt( l(t)) of balls radius t about '¥( t) for l a unit t>O
speed geodesic. These definitions make sense in greater generality: in
particular for simply connected manifolds of non-positive curvature. For
more details see [2], [8]. However there is no reason to believe that stable
manifolds for the canonical flows of these more general manifolds project
onto these horocycles.
Remark 4H(iii) For results about the non-triviality of the spectrum for the
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canonical flow when M * sn see [24a].
Remark 4HC1y) The characteristic exponents for the geodesic flow on the unit
sphere bundle in TM have been studied a lot [2 ], [106]. The results are
analogous for constant negative curvature: especially for dim M = 2 when the
bundle SOM can be identified with the sphere bundle. See also [105].
§5. Moment exponents
In this section we no longer require M ito be compact.
A. Consider a process Cxt : t ~ 0 on M and a process vt : t ~ 0 on some
space 8 with projection p : 8 -+ M such that p(vt(w)) = x1(w) for t ~ 0. If the
fibres p-1(x) of 8 are normed vector spaces for each x e M we can consider
for q e IR. Typical cases of interest are:
(i) 8 = M x IR and p the projection with Vt defined by
dVt/dt = V(Xt)Vt
(195)
(196)
for given v0 , for V: M-+ IR. We should then write Vp as vp(v0 ) etc. This is the
situation of the "Kac-functionals" studied extensively in [15], [54], [55]
especially in the non-compact case, i.e. the behaviour as t-+ oo of
I t V(x )ds 0 s
1/t log IE e (for v0 * O).
(ii) The analogue of (i) for 8 = M x IRn and V: M-+ n..(IRn; IRn)
(iii) p: 8-+ M the tangent bundle or a tensor bundle like APT*M with Vt defined
by a covariant equation
Dvt/ot = V(x1)vt (197)
where V(x) e n..(p-1(x), p-1(x)) for each x in M. One could equally well take
other vector bundles over M with a linear connection: this would then include
(ii) as the special case of the trivial bundle.
(iv) Vt = T x0
Ft(v0 ) where Ft(-,w) : t ~ 0, w e Q) is the flow of an S.D.E. on
M.
The last example is somewhat more complicated than the previous ones
since the equation for Vt is a stochastic differential equation in general. We
shall look in more detail at situations related to cases (i) and (iii) in the next
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chapter. Case (il) was investigated in [3]: there is the following general
result essentially taken from there as in [8]. In case (iv) it relates these
moment exponents to the Lyapunov exponents of the flow.
Proposttfon SA
Let v- and v _ be the random variables
v- = Tiiii1
11t log lvtl -+oo
and v- = lim.t-+oo 1/l log lvtl
Then
(i) q-+ vq is convex
(ii) q-+ 11q Vq is increasing
Also if IEiv-1 < oo and IEiv _I< oo
(iii) Vq;::: lim 11t IE lvtlq;::: q lEv_ q;::: 0
Vq2:QIEV_ q;:::O
d d -vql
0 ::s; IE v_ ::s; -vql
0 .
dq- q= dq+ q= (iv)
f.r:.2.Q1 [3] Part (1) comes from the convexity of q-+ log IE IZIQ for any random
variable Z and (ii) comes from the monotonicity of q-+ (IEIZIQ)ilq for q > 0 and
of q-+ (IE(11IZ!)-Q)- 11q for q < 0. Also by Jensen's inequality
11t log IE lvtlq ;::: IE 11t log lvtlq .. q IE 11t log lvtl
if log lvtl is integrable, so that (iii) and hence (iv) follows by Fatou's
lemma. II
8. To show one reason for studying the moment exponents let us go back to
the canonical flow on OM of a Riemannian manifold. Assume it is
stochastically complete so the solutions of the canonical S.D.E. ex1st for all
time. There is then the formal derivative flow which can be represented by
(s1, At) e IRn x Q.(n), as before, satisfying (121a,b) and (122a,b). Using the
notation of §4 set vt = UtSt· We can consider a 1-form cp on M as a section of
T*M or as cp: TM-+ IR with the restrictions fPx: T xM-+ IR linear. The element in • T xM dual to fPx wm be written fPx •
194358
Lemma 58
For a c2 1-form q> on M
t t q>(Vt) = q>(v0 ) + f Vq>(Usd8s)Vs + f q>(UsAsd8s)
0 0
1 t 2 . * + 2 J (trace V q>(Vs)ds - R1c(q>x, Vs))ds
0 s
(198)
Proof Coming back to the S(t,u)e and <SS(t,V)e notation of §98 Chapter II recall
that by Lemma 98 of Chapter II if J(t,V) = Tn(&S(t,V)) then J(-,V) is a Jacobi
field with Dlot J(t,V) = uro(V)e at t = o. Therefore, with 1(t) = nS(t,u)e,
dldt q>(TnoS(t,V)) = Vq>('Y(t))(J(t,V)) + q>(Diot J(t,V))
= Vq>(ue)(TnV) + q>(uro(V)e)
at t = 0; and, at t = 0,
d21dt2 q>(Tn&S(t,V)) = v2q>(ue,ue)(TnV) + 2Vq>(ue)(uro(V)e)
- q>(R(TnV,ue)ue).
Now for an orthonormal basis e1, ... ,en of IRn if f; = uei and S is the skew
adjoint operator uro(V)u-1
• Li Vq>(ue1)(uro(V)e;) = Li Vq>(f1)(Sf1) = Li (Vq> (fi),Sfi)
=-trace sv•q> =- trace(V.q>)S • =- Li ((Vq> )S fi, fi) =- L Vq>(Sf;)fi
=- L dq>(Sfi,fi)- L Vq>(fi)Sfi.
Thus (198) holds by Ito's formula. I I
The (de Rham-Hodge) Laplacian l:::.q> of a 1-form q> satisfies the
Weitzenbock formula
l:::.q> =trace v2q>- Ric(-,q>•) (199)
(with non-standard sign conventions), see Proposition 3D of Chapter V, below,
for the proof. The following result is discussed in [75], [43], [77].
Theorem 58 Suppose the family q>t: t ~ 0) of 1-forms on M satisfies:
(i) q>t is c2 on M and c1 in t, with the partial derivatives jointly
continuous,
(ii) oq>tlot = ~ t::.q>t t > o
195359
(iii) d<J>t = 0 t > 0
(iv) 'Pt is bounded uniformly in t e [O,T] each T > 0.
Assume M is stochastically complete and lvtl lies in L1 for each t where
Vt = ut st for so= uo-1v0 any frame u0 at x0 , and A0 = 0.
Then
!9t(v0 ) = IE<p 0 (vt) V0 E T x M, t ~ 0. 0
Proof Set 'Pt = IPT -t for 0 :S t :S T and apply the time dependent version of
Lemma 58 to 'Pt· I I
Note that (iv) holds automatically if M is compact as does the integrability
of lvtl. It is also true that d<p 0 = 0 implies d<pt = 0 when (ii) is satisfied, at
least for M compact. Furthermore, as we will see below, t::,<p = o implies d<p =
o forM compact. Thus
Corollary 58(1). If M is compact and li mt IE !vtl = 0 there are no -+00
harmonic 1- forms except 0. I I
Note that from the analogue of (170a) for constant curvature k if we
substitute klssl2 + 1Asl2 =owe see
1Eivtl2 = 1Eistl2 = e-3kt Ivai
and so the conditions of 5B(i) hold if k > 0. By (160) we have
t t iE lvtl2 = ls0 12 - f IE Ric (vs,Vs)ds + f IE 1Asl2 ds
0 0
so its hypotheses cannot hold if Ric (v,v) < o:lvl2, for all v, for some ex< 0. On
the other hand if the Ricci curvature is strictly positive everywhere Bochner's
theorem implies that there are no non-zero harmonic 1-forms. This will be
discussed in detail below.
C. For more about moment exponents and also their relationships with large
de vi ati on theory see [3 ], [ 4 ], [ 11 ], [ 13], [25], [ 45].
196360
CHAPTER IV. THE HEAT FLOW FOR DIFFERENTIAL FORMS AND
THE TOPOLOGY OF M.
§1. A Class of semigroups and their solutions.
A. Let p : 8 _. M be some tensor bundle over a Riemannian manifold M e.g. 8 =
TM, T*M, APTM, or a trivial bundle M x IRn, with induced inner product on each
Bx := p-1(x) (in fact any Riemannian vector bundle with a Riemannian
connection would do). For x e M suppose we have a linear map Jx: 8x _. Bx
depending measurably on x. Let x 1 : t ~ 0) be Brownian motion on M from the
point x0 : we will assume M is stochastically complete.
For v0 e Bx define the process vt; t ~ 0 over xt: t ~ 0 by 0
Dv1/ot = Jx1<v1) (200)
as in equation (47). Assuming J is bounded above (i.e. the map j defined below
is bounded above) the solution of (200) will exist for all time and
d/dt lv1!2 = 2 <Jx1<vt),vt>xt (201)
::; 2 j(xt) lvtl2 (202)
if j(x) = sup <Jxv,v) : v E Bx and lvl = 1.
Thus
l
J j(xs)ds
lvtl::; e 0 (203)
By a c2,1 section fPt : t ~ 0 of B* we mean a time dependent section of
the dual bundle to B: so fPt x E ll...(Bx;IR) for x e M and the map <p : 8 x [O,oo) 4 IR , . given by (v,t) 4 !flt(Y) has two partial derivatives in the first variable and one
in t, all of them continuous. The following can be considered as a uniqueness
result:
Proposition 1A. Suppose fPt : t ~ 0 is a c2,1 section of 8* such that
ocpt/ot = ~ Trace v2<pt + J*(cpt) (204)
with fPt bounded (i.e. CI~Pt xI: x eM bounded) uniformly on each 0::; t::; T, , forT > 0 • Then if J is bounded above, and v0 e Bx some x0 e M
0
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C9t(Vo) =IE C9o(Vt) (205)
where (vt: t ~ 0 is the solution to (200).
Proof: To interpret (200) we can suppose Xt = n(ut) for (ut : t > 0) a solution
to the canonical S.D.E. on OM. For S(t,u)e as before (e.g. §SA Chapter II) and
6 t = nS(t,u)e suppose
Dlot vt(e) = J.n<vt(e))
with v0 (e) = w 0 some w 0 e s60
• Then for tp : 8-+ IR of class c2 and linear on
the fibres, if x = n(u) = lo
dldt tp(vt(e)) = Vtp('Yt)Cvt(e)) + tp(J.'tt(vt(e)))
and, at t = 0,
d21dt2 tp(Vt(e))
(206)
= V2tp(ue,ue)(w 0 ) + 2Vtp(ue)(Jx(w 0 )) + tp(V J(ue)(w0 )) + Jx(Jx(W0 )) (207)
if J is differentiable. At first sight it is not obvious how to interpret this to
obtain the Ito formula for tp(vt) using Proposition 3A of Chapter I. In fact our
system does fit into that result but with Zt = (Bt,t), and as a system on OM x
Bx , namely 0
dUt= X( Ut) o dBt
dwt = llt- 1 J11 (ut) UltWt)dt
where I It is parallel translation of the tensors along (n(us): 0 :$ s :$ t: (this
is p(u 0 ut-1) for a suitable representation p of O(n) on Bx ). Thus the terms 0
above without adequate e's will have a 'dt' in Ito's formula and the terms
involving VJ will have a 'dt dsi 1• and so not appear. The assumption of
differentiability of J is therefore not needed (remember the global Ito formula
depends on local formulae, the way we are working out its coefficients is just
formalism: a method of obtaining a formula whose coefficients have geometric
content). Thus
t t
tp(Vt) = tp(Vo) +I Vtp(UsdBs)(Vs) +I (~ trace v2tp(Vs) + tp(Jx (Vs)))ds. (208) 0 0 s
Alternatively this can be derived from the Stratonovich equation
198362
t t
ttJ(Vt) = ttJ(v0 ) + J VttJ(Us o dBs)(vs) + J ttJ(Jx (vs))ds 0 0 s
(209)
The result follows by applying the time dependent form of (208) with 'tit =
<l'T -t where T > 0. I 1
Versions of (208) for more general systems than (200) are given in [14].
B. After a 'Feynman-Kac type' formula here is a 'Girsanov-Cameron-Martin'
formula. Let A and Z be c1 vector fields on M. Let (xt: t 2:: 0 denote Brownian
motion on M from x0 with drift A , assumed non-explosive, and (ut : t 2:: 0 its
horizontal lift to OM: so we can take it that -dUt = X(u1) o dBt +A (ut)dt (210)
-for (X,B) the canonical S.O.S. on OM and A the horizontal lift of A, with n(ut) =
Xt· Let Mt be the process on IR given by M0 = 1 and
dMt = Mt<Z(xt)•ut o dBt)- ~ Mt div Z(Xt) + IZ(Xt)l2dt
so
t t
Mt = expcJ (Z(xs),usodBs> - ~ J (div Z(Xs) + IZ(xs)l2 ds 0 0
In the more familiar Ito formalism
dMt = Mt(Z(xt), Ut dBt>xt
and
t t
Mt = expJ (Z(xs),us dBs> - ~ J IZ(xs)l2ds 0 0
(211)
(212)
(213)
(214)
Proposition 18 Suppose <pt : t 2:: 0 is a c2,1 section of B* such that
o<; 11at =~trace v2<t>t + V<pt(A) + V<p1(Z) + J*(<;t) (215)
and <l't is bounded uniformly on 0 ~ t ~ T for each T > 0 . Then with the
assumptions and notation above, if also the process with generator ~6 +A+ Z
is complete and if J is bounded above
<t>t<vo) = IEMt<t>o<vt)
for each x0 in M and v0 E Bx0
where vt : t 2:: 0 satisfies the covariant
equation along the paths of x1 : t 2:: 0
199363
(216)
Proof. First consider the case 8 = M x IR with p the projection and J = 0. This
is the classical theorem: for cp: M.,. IR which is c2 and bounded the Ito formula
for Mtcp<xt) shows Qtcp defined by Qtcp(x0 ) = IEMtcp<xt) is a minimal semigroup on
L00 with differential generator ~6 +A+ Z; there is a unique such semi-group - -so the change of probability to IP with IP = MTIP (on paths w restricted to 0::;
t ::; T) is a change to a probability measure and under IP- the process x1 : 0 ::;
t ::; T has generator ~ 6 + A + Z.
Now Proposition 1A extends with essentially the same proof to the case
where the process Cxt: t 2:: 0 has a drift. Applying this to (x1 : t 2:: 0 under the
probability IP- gives (215). II
Note:
(i) Under the completeness conditions (Mt : t 2:: 0 is a martingale.
(ii) we can allow A and Z to be time dependent provided their sum A+ Z is not.
C. A case which we will be particularly interested in is 8 = TM and J(v) = Ric(v,-)•. From the Weitzenbock formula (199) proved later (Chapter V, §3),
and elliptic regularity which shows that solutions to the heat equation for
forms are c2,1 (in fact C00 ) we have from Proposition 1A and Yau's result on
the stochastic completeness of M when M is complete with Ricci curvature
bounded below:
Theorem 1C. If M is complete with Ricci curvature bounded below then
any solution cpt : t 2:: 0 to the heat equation for 1-forms
acptlot = ~ 6cpt
with 97t uniformly bounded on compact intervals [O,T] of IRis given by
ct't<vo) = IEcpo(Vt)
where Cvt: t 2:: 0 satisfies the covariant equation along Brownian paths
Dvtlot = - ~ Ri c(vt,- )•. II
We will discuss the analogous situation for p-forms p > 1 later.
§2. The top of the spectrum of 6
A. For complete Mit is a standard result that 6 is essentially self-adjoint on
the space of C00 functions with compact support (as is trace v2 acting on
sections of 8* as in §1 and the de Rham-Hodge Laplacian on forms, [91]).
Since, for f of compact support,
200364
J fM=- f<Vf,Vf):sO, M M
b. is non-positive and so there is a semi-group induced on l2(M) by ~b. which
we will write e~tb.: t 2: 0). This semigroup restricted to L00 n l2 extends to
a contraction semi group on L 00 (M), e.g. see [86] p.209. By elliptic regularity
and the simplest case of Proposition 1A, this implies that
e~tb. f(x 0 ) = !Ef(xt) (217)
for f e L 00 n L2. We will not distinguish between b. and its (self adjoint)
closure.
There is the heat kernel Pt(x,y) fort> 0 and x,y e M. It satisfies
Pt(x,y) = lim i~
()()
D· Pt T (x,y) (218)
where D;i=t is an increasing sequence of bounded domains in M with smooth
boundaries whose union is M and where Pt0
i(x,y) denotes the heat kernel in Di
with Dirichlet boundary conditions. Equation (218) holds because the
corresponding result holds for the transition probabilities of Brownian motion
on M and the Brownian motions in Di killed on the boundary.
For an incomplete manifold (218) can be taken as the definition of Pt(x,y),
each Di having compact closure.
B. Si nee b. is a negative operator :>-. 0 := sup :A. e Spec b. :s 0. When M is
compact or has finite volume :>-. 0 = 0 since the constants lie in L.2. There are
various characterizations of :A. 0 (M) e.g. see [93]: in particular
:A. 0 (M) =- inf J IV<pl2 I J l<pl2: <pis C00 with compact support). M M
()()
Let D;h=t be an exhaustion of M by pre-compact domains with smooth
boundaries as before. The spectrum of the Laplacian with Dirichlet boundary
conditions for functions on Di is discrete. Let :>-. 0 (0;) be the first eigenvalue,
so :A. 0 (Di) < 0. Then
201365
A. 0 (Di) = -inf U0
. IV<pl2/ f0
.1<pl2: <pis coo with 1 1
compact support in Di
00 e.g. see (28]. Thus A. 0 (Di)i=l is increasing and
A. 0 (M) = lim A. 0 (o1). 1-+00
It is shown in [31], see also [93], that if p E o 1 and hi : Di -+ IR satisfies
. . . . n hl(p) = 1 and t.hl = A. 0 (Di)h1 then on any compact set in M the sequence (hl)i=l
has a uniformly convergent subsequence giving a limit h : M -+ IR which is
positive and satisfies t.h = A. 0 (M)h. A. smooth function h is a A.-harmonic
function if t.h = A.h. A basic result [93] is (for non-compact M):
There are positive A.-harmonic functions if and only if A.~ >-o(M) •
Note that for A. * >-o such functions cannot be in L2.
C. The Green's region consists of those A. with
a>
g>-(x,y) = ~ J e-~ A. t p1(x,y)dt < oo 0
for all x,y with x * y. From functional analysis if A. > >-. 0 then A. lies in the
Green's region. See [6], [93]. On the other hand if A. < A. 0 (M) then A. < A. 0 (D;)
for some Di. Writing A.i for A. 0 (Di) and t,i for the Dirichlet Laplacian for oi, if
A. were in the Green's region this would imply from
• 1"\it 1tAi • h1(x) = e-2"' e2 u h1(x)
that
t t
hi(x) = 1/t Jhi(x)ds = 1/t J e-~ >-.is (Jp0
i (x,y)hi(y)dy)ds 0 0 M S
t
:S 1/t J e -~ >-s<JD. Ps(x,y)h i (y)dy)ds 0 1
202366
t
= JD· (1/t J e -~A.s Ps(x,y)hi(y)ds)dy 1 0
-+ 0 as t -+ oo.
Thus [93],the Green's region consists of [A. 0 (M),oo) or (A. 0 (M),oo).
Given a positive A.-harmonic function h:M-+ IR (> O) we can h-transform
Brownian motion, and the heat semigroup Pt. For this define, for measurable
f : M-+ IR, the function phtf:M-+ IR by
pht f(x) = 1/h(x) e-A.t/2 Pt(hf)(x) (219)
when it exists. This gives a semigroup with differential generator Ah where
Ahf(x) = ~M(x) + (V logh(x), Vf(x)>x (220)
so the corresponding Markov process is Brownian motion with drift Vlogh; this
is the h-transformed Brownian motion on M. The fundamental solution is
given by
Pth(x,y) := 1/h(x) e-A.t/2 Pt(x,y)h(y) (221)
From general results about transience and the existence of Green's operators
in [6 ], or from [93], we see A. is in the Green's region if and only if the h
transformed Brownian motion is transient.
D. The following result from [93] will be very useful, as will its method of
proof which comes from [46. See also [85]
Proposition 2P If A. belongs to the Green's region then for every x0 eM
and compact set K of M, if xt : t 2: 0 is Brownian motion from x0
1. -(A./ 2)t IP K) - 0 rrn e Xt E - • (222) t-+OO
Proof Choose a A.-harmonic function h : M-+ IR(> 0). Let (Yt : t 2: 0) be the h
transformed Brownian motion from x0 • It is transient and so lim IE XK(Yt) = 0 t-+OO
i.e. lim pht(XK)(Xo) = 0 t-+OO
By definition of pht this means lim e-~A.t lEh(xt)XKCxt) = 0 t-+OO
which gives (222) since his bounded away from 0 on K.
203367
Corollary 20 ForK compact and (xt : t 2: 0 Brownian motion on M
lim 11t log IP(Xt E K :$ ~ A. 0M) (223) t-+OO
Proof By (222) if :>... > :>... 0 (M) then lim 11t log IPXt e K :$ ~ :>.... I I t-+00
§3. Bochner theorems for L2 harmon1c forms
A. There is a wide literature in differential geometry relating curvature
conditions on M to the existence of functions, forms or tensors, of particular
kinds e.g. see [14], of these Bochner's theorem has been particularly
important: the simplest case of it says that if the Ricci curvature of M is
positive definite and M is compact then there are no harmonic 1-forms. The
importance is because of Hodge's theorem which states, in particular, that the
dimension of the space of harmonic 1-forms is the 1st Betti number of M when
M is compact i.e. it is the dimension of the first cohomology group H1 (M;IR) of
M with real coefficients. This case of Bochner's theorem is very simple from
what we have done in Theorem 1C with equation (202). It is also almost
immediate functional analytically from Wei tzenbock's formula and the
following. Let q;' n_ (TM; T*M) be the space of C00 sections of the tensor
bundle over M whose fibres consists of R.(T xM; T x*M) for x in M, with
corresponding notation for other spaces of sections.
lemma 3A Let V* : q;' R.(TM; T*M) -+ q;'(T*M) be the formal adjoint of
V : q;' T*M-+ q;' n.(TM; T*M). Then for <p e q;' R.(TM;T*M),
(V*r.p)x =-trace (Vr.p)x =- ~=t Vr.p(ei)(ei)
where e 1, ... ,en is an orthonormal base for T xM·
(224)
Proof Given r.p e q;' ll(TM; T*M) and IV e q;'CT*M) there is the one-form
given by v-+ <r.px(v),I.Jix> for veT xM with corresponding vector field x-+
<r.px(-), IJ'x)•. By the divergence theorem
204368
0 = J div (<p_(-), tp_)• M
J trace V(<p_(-), tp_)• M
=J M
L (V<p(ei)(ei),tp) + (<p(ei), Vtp(e;). i
But ~(<p(ei), Vtp(ei)) = (<p,Vtp) and so (224) follows. I I 1
From Lemma 3A the Weitzenbock formula (199) can be written
t:.<p = -V*V<p- Ric(<p*,-) (225)
for a 1-form <p. Thus if <p has compact support and is smooth
(6<p,<p)L2 =- (V<p, V<p)L2 - J Ric (<p•, <p•) (226) M
from which Bochner's theorem for compact M with Ric(v,v) > 0 for all v * 0
follows. In fact it clearly extends to the case of non-compact M if we consider
only L2 forms <p (and are careful about the existence of V<p in L2 if t:.<p = 0).
See [39], for example, for generalities about L2 harmon1c forms etc.
B. The follow1ng is an 1mproved version of the L2 Bochner theorem, taken
from [46]. The case of strict equality can also be fairly easily obtained
analytically, using the method of domination of semi-groups a direct parallel
of the proof given here) as 1n [40] §4. For x eM set .Bk(x) = inf Ric(v,v): lvl =
1, vET xM.
Theorem 3B [46]. Assume M is complete with B.k(x) 2:: A. 0 (M) for all
x eM and also either:
(i) B.k (x) > A. 0 (M) for some x e M or
(ii) A. 0 (M) is in the Green's region.
Then there are no L2 harmonic 1-forms except 0.
Before giving the proof we need a few more facts about the Laplacian on
forms. It is usually deflned (although with the opposite sign) on smooth forms
by
!;, = -(d& + &d) (227)
where d is exterior differentiation and & is the L2 adjoint of d. These are
205369
discussed in more detail later, and the Weitzenoock formula proved. For the
moment simply observe that if tp is a sufficiently regular one-form then
(6tp,tp)L2 = -(dtp,dtp)L2 - (otp, otp\2 (228)
since dd = 0. Thus 6 is a negative operator. It is essentially self adjoint on
c; and so there is a naturally defined semigroup (e~t6: t > 0). By regularity
theory, if the Ricci curvature is bounded below, Theorem 1C identifies e~t61P with PtfP• when tp is in L2 n L00 • for Pttp(v 0 ) = IEtp(Vt) with the notation of
Theorem 1C. See [91a] and the discussion in [46].
When we wish to distinguish between the Laplacian on forms and on
functions we will use 61 and 6o, with pit and pot for the corresponding
probabilistically defined semigroup.
Proof of Theorem 38. Suppose there is a non-zero L2 harmonic 1-form tp 0 •
Choose a smooth Jl : M-+ 1R(2: 0) with support in some compact set K such that tp
:= Jl'Po is not identically zero. The space of L2 harmonic 1-forms is closed in
L2 (it is (tp e L2: et6tp = tp for all t > 0). Let H be the projection in L2 onto it.
Then Htp * 0 since
(Htp, 1Po>L2 = (fP,fPo>L2 = J J.l<fPo,fPo> > 0. M
By abstract operator theory e~t6tp-+ Htp in L2 as t-+ oo. A subsequence
therefore converges almost surely on M, say on some subset M0 of M. Choose
x0 E M0 • Set v0 = (Htp)•x E T x M. Then 0 0
(229)
Set C = inf Ric(x). For Dvt/ot = - ~ Ric(vt,-)• along Brownian paths, by
Theorem 1C and estimate (202), equation (229) gives
(230)
which implies C is not in the Green's region by Proposition 20. By assumption
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C :::: :>.. 0 (M). Therefore C = :>.. 0 (M) and :>.. 0 (M) is not in the Green's region i.e. (ii)
does not hold.
To contradict (i) assume it holds and take a :>.. 0 -harmonic function h: M-+
IR(> 0) for :>.. 0 = :>.. 0 (M) = C. Let z1 = xht• the h-transformed Brownian motion
starting at x0 • Its generator is ~6 + Vlogh. Since :>.. 0 is not in the Green's
region Zt: t:::: 0 is recurrent and hence complete and we can apply
the Girsanov theorem, Proposition 18, toP 1t(c.p) to get
e-~t6 c.p(vo) =IE Mt c.p(vt)
where vt: t:::: 0 satisfies Dv1;at =- ~ Ric(vt,-)• along the paths of
Czt: t :::: 0 and
t t
Mt = exp (- J <Vlogh(zs), UsodBs>- ~ J ( -6logh(zs)ds + IV log h(zs)!2)ds 0 0
t t
= exp -log h(zt) + 1ogh(x0 ) + ~ J 1Vlogh(zs)l2ds+ ~ J 6 logh(zs)ds 0 0
(240)
since 6logh = h-16h- 1Vloghl2 = :>.. 0 - IVloghF. Here u1 refers to the horizontal
lift of (Zt : 0 '$ t < oo.
Take a bounded open set V of M with ill (x) > o + C for x in V, some o > 0.
Let At= s E [O,t): Zs 1$ V and Bt = s E [O,t]: Zs e V. Then
t
exp - ~ J ill (zs)ds '$ exp -~ CIAtl- He + o)IBtD 0
'$ ex p (- ~ C t - ~ o IB t I
where IAtl• IBtl denote the Lebesgue measures of the random sets At• Bt.
Consequently
0 < lim IE M1c.p(vt) t-+OO
'$ lc.piL 00 h(x0 )(i nf h(x) : x e K )-1 lim IE exp (- ~ o!B 11)
since :>.. 0 = C. However IBtl-+ oo as t-+ oo almost surely by the recurrence of
zt: 0 '$ t < oo (e.g. see [5] proof of Lemma 1), so this is impossible. Thus (i)
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cannot hold either. I I
Corollary 38 If M is complete and has a non-trivial L2 harmonic one
form then
A. 0 (M) 2: inf ...B.k(x) X
(241)
with strict inequality if either Ric is non-constant or A. 0 (M) is in the Green's
reason. II
This compares with Cheng's estimate for A. 0 when inf B..i.k.(x) < 0, [30]:
A. 0 (M) 2: 114 (n-1) inf x Bi.£x) (242)
and improves it for n > 5 given some non-trivial L2 harmonic 1-form
These results are discussed in relation to quotients of hyperbolic spaces
in [46].
Remark 38
(l) Corresponding results for p-forms can be proved in the same way given the
Weitzenbock formula for the Laplacian on p-forms (see below), and similarly
for the Dirac operator, [46]. The discussion in §1 shows how to formulate a
general theorem.
(ii) For compact manifolds Theorem 38 reduces to the classical Bochner
theorem. Note that the flat torus s1 X s1 has Ao(M) = .B.i£(x) = 0 for all X but
has harmonic 1-forms, e.g. d91 and d92 where (a1, a2) parametrize s1x s1 by
angle. Thus some additional conditions like (i) or (ii) are needed.
§4. de Rham cohomology, Hodge theory, and cohomology with
compact support.
A. Let AP be the space of C00 p-forms on M. (See Chapter V, §3.) Exterior
differentiation d gives a map
d: AP-+ AP+1
and the p-th de Rham cohomology group HP(M;IR) is defined by
p p+l HP(M;IR) = ker(d: A -+A )
Im(d: Ap-t-+ AP) (243)
It is a classical result that it is isomorphic to any of the standard cohomology
groups with real coefficients (e.g. simplicial or singular). The de Rham-Hodge
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Laplacian on p-forms, 6, or t:,P to be precise, is given by
t:.P = -(do + oct) (244)
where o is the formal adjoint of din the L2 sense. On the space Ab of p-forms
with compact support it is known to be essentially self-adjoint, e.g. see [91],
and so we can take its closure which will be self-adjoint. This will still be
written as t:,P. There is then the corresponding heat semigroup e~tt:.P acting
on the space L 2 AP of L 2 p-forms since t:.P is non-negative by the same
argument as for t:,1 ; see equation (228).
Let H = HP: L2AP-+ L2 AP be the projection onto the space of harmonic
p-forms. Then, as before, e~tt:.-+ H strongly on L2AP. For rp € L2AP set
00
Grp = J (e~tt:.- H)rp dt. 0
Then, leaving aside rigour for the moment,
00 00
t:.Grp = J t:.e~tt:.rp dt = J o/ot(e~tl::.rp)dt = Hrp- rp. 0 0
Thus we have the decomposition for rp e L2AP
rp = -t:.Gcp + Hcp (245)
From (244) we may believe the Hodge decomposition theorem, at least for
compact manifolds (when t:.P has discrete spectrum): any rp e AP has a
decomposition into three orthogonal summands
rp = Hcp + dcx + o j3 (246)
for ex e L2Ap-1 and j3 e L2Ap+1. In particular if dcp = 0 then cp = Hcp + dcx
since (cp, &J3\ 2 = (dcp,j3)L 2 = 0. Thus we have Hodge's theorem: every
cohomology class has a unique harmonic representative. In particular the p
th Betti-number J3p
J3p := dim HP(M;IR) = dim (space of harmonic p-forms
when M is compact.
For non-compact manifolds the heat equation method outlined above was
used by Gaffney to get a version of Hodge's theorem [53]; see also [37], [38].
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B. The operators d and o can be restricted to the space A b to give the
cohomology groups HPK(M;IR) of M with compact support by the analogue of
(243). There is a natural inclusion
i p : HPK(M;IR) -+ HP(M;IR)
whose kernel has elements [<p]K represented by forms <p e A b with <p = dex
some ex in Ab. If <p' = dex' is also in Ab then [<p']K = [<p]K if and only if ex= ex'
outside of some compact set.
In the special case p = 1 this gives a linear surjection
E(M) d. -+
where E(M) is the quotient of the vector space of bounded ceo functions
f : M-+ IR with df e A 1 0 by the space A 0 0 of ceo functions with compact
support. The kernel of this map consists of [f] : f is constant and so we have
an exact sequence
The set of ends EndM of M is the projective limit of the inverse system
whose terms are the sets of connected components of M-K as K ranges over all
compact subsets of M, directed by inclusion. Thus an end is an indexed set EK
: K compact) such that EK is a component of M-K and if K c K' then EK• c EK.
They can be thought of as the components of Mat infinity and we will say that a
continuous path Cl : (O,eo)-+ M goes out to infinity through the end E or 'lim Cl(t) t-+OO
= E' if for each compact K there exists tK with Cl(t) e SK for t > tK.
There is a natural map j: S(M)-+ IREnd(M) given by j([f])(S) = limKf(x): x e
EK) which is clearly injective. Thus M is connected at infinity (i.e. has a
unique end) iff S(M) = IR.
C. A simple example is that of the cylinder M = s1 x IR. This has two ends, so
S(M) ~ IR ffi !R. If it is parametrized by (e,x) where e represents the angle,
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there is the one form de which determines the generator of H1(M;IR), which is
isomorphic to IR since M is homotopy equivalent to s1. Now [de] clearly does
not lie in the image of i 1 (otherwise it would have to be exact outside a
compact set i.e. equal to df , for some function f , on the complement of some
compact set). Therefore H1 0 (M;IR) ~ IR generated by [df] where f(x) = +1 for
x > 1 and f(x) = -1 for x < -1.
D. The relevance of these concepts to L2 harmonic form theory and Brownian
motion comes from the Hodge decomposition for cp e A b, [37], with dcp = 0,
which gives
cp = Hcp + do:
foro: e AP-1. From this we see that if there are no non-trivial L2
harmonic p- forms then the map
ip: HP 0 (M;IR)-+ HP(M;IR)
is identically zero. The point is that the latter is a topological condition
independent of the Riemannian metric (and in fact even of the differentiable
structure). From Corollary 38 we can now say that if i 1 is not identically zero
then :>.. 0 (M) ~ inf B.ll.(x); see [46] for some examples. This relationship X
between L2 harmonic forms and i 1 was exploited by Yau [10] for complete
manifolds with non-negative Ricci curvature. Using properties of such
manifolds (in particular the Gromoll-Cheeger splitting theorem) he was able to
give conditions for the vanishing of ker i 1 and hence for H10 (M;IR) itself rather
than just its image in H1(M;IR). Analogous results for B.k(x) > :>.. 0(M) are given
in [46], but different methods are needed, and additional conditions of 'bounded
geometry' appear to be needed for these methods to work. One such result is
described next.
§5. Brownian motion and the components of M at 1nf1nlty
A. It is shown in [ 46] that if M has bounded secti anal curvatures and a
positive injectivity radius (i.e. there exists r > 0 such that expx: T xM-+ M is a
diffeomorphism of the ball of radius r about 0 onto some open set in M for each
x eM) e.g. if M covers a compact manifold, then Ricx) > :>.. 0 (M) for all x implies
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that M is connected at infinity, and so by the previous discussion H10 (M;IR) = 0.
The proof uses some Green's function estimates by Ancona. Here we will
discuss another result from (46] which is very similar but has a slightly
different emphasis.
Theorem 58 (46]. Let M0 be a compact Riemannian manifold. Assume there
exist € > 0 with
J Ri cx)f(x)2dx > - J IVf(x)Fdx + €
~ ~ (247)
for all C00 functions on M0 with lfiL 2 = 1. Then every covering manifold M
of M0 is connected at infinity (i.e. has at most one end). Moreover H10 (M;IR)
= 0 and M, with covering Riemannian structure, has no non-trivial harmonic
1-forms in L2.
The condition on M0 is precisely the condition that the top of the
spectrum of ).-Ric is negative.
Proof: To show connectness at infinity take f : M -+ IR smooth and bounded
with df having compact support. It suffices to show that such a function f is
constant outside of any sufficiently large compact set.
Let u1, u2 be among the unbounded components of M-supp(df). For p : M
-+M 0 the covering map, take x0 eM and choose xi: 1 in u1 and yi: 1 in U2
with xi-+ oo and Yi -+ oo and p(x;) = x0 , p(y;) = x0 for each i. (We can assume M
non-compact of course.) Take a finite set of generators for the fundamental
group TT 1 (M 0 ,x 0 ) and let g1, ... ,gr denote them together with their inverses.
Choose smooth loops 'h•····lr at x0 , with each lj in the class gi and with tlj(s)l = 1 for all s and j e.g. the lj could be geodesics. Take a shortest path ex from xi
to xi+l· Then p o ex is a loop at x0 and we can write its homotopy class [p o ex]
as a product gil ••• gis for some s • Lift lis to a path in M starting from
xi and lift the other corresponding paths in turn to start where the previous
lift ended and so give a continuous piecewise c1 path from x1 to xi+1.
Do the same for x1 to y1 and Yi to Yi+l for each i. Let 6: (-oo,oo)-+ M be
the curve obtai ned from the union of these lifts: it is piecewise C 1 and
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satisfies lcrsl = 1 for all s where cr(s) is defined and there exists T e IR such
that o(s) e u 1 if s <-T and o(s) e u2 if s > T.
Let ll be the set of all open subsets of M on which p is injective. For
compact K in M let c(K) be the minimum number of elements in ll needed to
cover K, and suppose U1•···•Uc are in ll and cover K. Then
00 00
J XK (6(s))ds ::s ! J Xu.(O(s))ds . 1 J
-00 )= -00
Now the intersection of the curve o with Uj decomposes into portions P 1, ... ,P r
where each Pi consists of pieces which come from lifts of li· Since p is
injective on Uj it maps each Pi injectively into li· Therefore
Joo XU· (6(s))ds ::s t J00
(Xp.(6(s)))ds J 1=1 1
-oo -oo
where .t(li) is the length of li· Thus for any compact Kin M
Joo XK(O(s))ds ::s c(K) t .t(l1·). 1= 1
(248) -00
We can now show that J P:(df) exists for t ~ 0 and converges to zero as t-+ (J
oo. To do this take a smooth flow of diffeomorphisms on M0 of Brownian
motions, F0 t(-,w) : M0 -+ M0 for t > 0, w e Q, e.g. a gradient Brownian flow
as in Chapter III. This is possible because M0 is compact. It lifts to a
smooth Brownian flow of diffeomorphisms Ft(-,w): M-+ M, t ~ 0, w e Q. Set
K = supp (df). Then by (202) and Theorem 1C
00
1J P:(df) I ::s J IP 1t(df)l(os)ds (J -00
213
00
:i J -co
377
t
IE(!dfi(Ft(<'(s)))exp- J Bk..(Fr(d(s))dr)ds 0
co t
:i ldfiL 00 J IE(XK(Ft(d(s)))exp- J ~(Fr(<'(s))dr)ds -co 0
where K = supp (df).
(249)
To avoid worrying about regularity properties stemming from the possible
lack of smoothness of B.1£ choose a smooth map p 0 : M0 -+ !R with Ric(x) 2: p 0 (x)
for all x and such that condition (247) holds for .B.i£ replaced by p 0 • Let v
denote the top of the spectrum of 6.-p 0 on M0 • The revised condition (247)
implies that v < 0. From Perron-Frobenius theory (e.g. see
[86]) there is a strictly positive h0 : M0 -+ !R (> 0) with
6.h0 (x) - p 0 (x)h0 (x) = vh0 (x)
for x e M0 •
Let h = h0 o p : M-+ IR(> O) and p = p0 o p • There is a flow for the h0 -
transformed Brownian motion on M0 and a lift of it to a flow Fht, t 2: 0, say, on
M of h-transformed Brownian motions. By the Girsanov theorem using the
analogous computation as that which led to (240), from (249) we get
co
1J P:(df) I :i ldfiL 00 J (IE XK(Fht(<'s))e~vt h(<'s)/(h(Fht(<'s))) CJ -co
co
:i const. e~vt J IE XKt(<'s)ds -co
t
exp - J (B.k- p)(Fhr(<'s))dr)ds 0
where Kt is the random compact set (Fht)-1(K). However c(Kt) = c(K) since ll is
invariant under those diffeomorphisms of M which cover diffeomorphisms of
M0 • Therefore
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r
I J P; (df) I :S const. e~vt c(K) ~~(li) (J
-+ 0 as t -+ oo.
Thus
On the other hand, by the compactness of M0 , all the curvature
tensors and their covariant derivatives are bounded on M0 , so by Theorem 58
of Chapter III, pis b.(df) =pis d(M) = dPos M (or alternatively by [52],
pis 6 df = c,p1s df = -(d& + M)P1s df = dPos M). Therefore
R t
J (Pt df- df) = lim J d J P0 s(M)ds a("t)d't CJ R-+00 -R 0
t t
=lim J P0 s(b.f)(<'(R))ds- J P0 s(M)(<'(-R))ds R-+oo 0 0
= 0
by dominated convergence and the 'C 0 -property' of the semigroup P 0 t : t 2: 0.
This last property says that Pt(g)(x)-+ 0 as x -+ oo for each t whenever g is
continuous with g(x)-+ 0 as x-+ oo. It was shown by Yau to hold for complete
manifolds with Ricci curvature bounded below, e.g. see [iOO]. Alternatively it
follows rather easily from the existence of a Brownian flow of
diffeomorphisms [46].
Thus
0 =lim J P; df = J df =lim f(<'(R))- lim f(<'(-R)), t-+00 CJ CJ R-+oo R-+oo
and so f]Ui = f1U2, proving the first part of the theorem.
Next we observe that M has no non-trivial harmonic forms in L 2 by
arguing by contradiction as in the proof of Theorem 38 but using the h
transform this time for h as above. The triviality of H10 (M;IR) follows from
the discussion in §4. I I
215379
CHAPTER V HEAT KERNELS: ELEMENTARY FORMULAE. INEQUALITIES. AND SHORT
TIME BEHAVIOUR
§1. The elementary formula for the heat kernel for functions.
A. We will be following [47] and [81] fairly closely in this section. For a
Riemannian manifold M and continuous V : M ~ IR, bounded above there is a
continuous map
(t,x,y) ~ Pt(x,y)
(IR > 0) X M X M ~ IR
such that the minimal semigroup (Pt: t ~ 0 for ~6 + V has
Ptf(x) = f M Pt(x,y)f(y)dy t > 0 (250)
for bounded measurable f. This is the fundamental solution. If
Di : i = 1 to oo is an increasing sequence of domains exhausting M, with
smooth boundaries, and if pti(x,y) denotes the fundamental solution to the
equation
oft/at= ~Mt + Vft
on Di with Dirichlet boundary conditions then
D· Pt(x,y) = lim Pt 1 (x,y)
i-+OO
(251)
(252)
and the right hand side is an increasing limit. This is clear from the
Feynman-Kac formula, or alternatively we can define Pt(x,y) by (252), with
compact Di, and then Pt by (250). In either case in order to obtain an
expression for Pt(x,y) it will be enough to find one for the fundamental
solutions on each Di, with Di compact, and then take the limit.
B. To obtain exact formulae for these fundamental solutions we will need
some rather strong conditions on the domains, and on M. However these
conditions will turn out to be irrelevant when the asymptotic behaviour of
pt(x,y) as t.!. 0 is being considered, at least for generic x andy and complete M.
To describe these conditions we need to look in slightly more detail at the
exponential map.
First suppose M is complete. For p EM let
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U(p) = v e T pM: d(exppv,p) = lvl
and let oU(p) be its boundary and U0 (p) its interior. The following facts can be
found in [16], [29], [63]: The image Cut(p) of oU(p) is a closed subset of M
known as the cut locus of p, moreover:
(a) U0 (p) is star shaped from the origin in T pM
(b) expp maps U0 (p) diffeomorphically onto the open subset M- Cut(p) of M.
Example 1: M = s1. Here the exponential map wraps T ps1::::: IR around s1 as a
covering map (it is locally a diffeomorphism), and Cut(p) is the point antipodal
to p.
Example 2. M = sn for n > 1. This is quite different from M = s1 since the
exponential map is no longer a local diffeomorphism: it maps the whole
sphere radius n to the antipodal point of p. Again Cut(p) is this antipodal point.
Example 3. Real projective space: M = IRIP(n). This is the quotient space of sn
under the equivalence relation x - y if x is antipodal to y. It is given the
differentiable structure and Riemannian metric which makes the projection p:
sn-+ IRIP(n) a Riemannian covering. If x e IRIP(n) corresponds to the North (and
therefore the South) pole of sn then Cut(x) is the image under p of the equator,
a copy of sn-1. Thus Cut(x) is a submani fold, isometric to IRIP(n-1), in IRIP(n).
It has co-dimension one and so will almost surely be hit by Brownian paths
from x in IRIP(n).
Example 4. M = Hn, hyperbolic space. In §4G of Chapter III we saw that there
are global exponential co-ordinates about a general point p. Thus Cut(p) = f!J.
Example 5. Complete manifolds with non-positive sectional curvatures
("Cartan-Hadamard manifolds"). The Cartan-Hadamard theorem e.g. [65], [79]
states that for such manifolds (e.g. M = s1) each exponential map expp: T pM-+
M is a covering map. In particular it is a local diffeomorphism. (To prove this
see Exercise 1A below.) It follows that if M is simply connected then expp is a
diffeomorphism and Cut(p) = f!J for each pin M.
When Cut(p) = f!J, so that there exists a global exponential chart about p, the
point p is said to be a pole of M. If so, M is diffeomorphic to IRn and so is
essentially IRn with a different metric. The images under expp: T pM-+ M of a
point v such that the derivative Tv expp of expp at v is singular is called a
conjugate point of p along the geodesic expp tv : 0 ~ t < oo, and v itself is
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said to be conjugate top in T pM·
Exercise 1A Show that the derivative of expp at v in the direction w is given
by
Tv expp(w) = J1
where Jt: 0 :S t :S 1 is a vector field along exp tv: 0 :S t :S 1 with J0 (0) = 0
and DJt/otlt=O = w. (Here Tv T pM is identified with T pM using the vector space
structure of T pM). Hint: look at the proof of Lemma 98 of Chapter III. Thus v
is conjugate top in T pM if and only if there is a non-trivial Jacobi field along
exp tv : 0 :S t :S 1 which vanishes at t = 0 and at t = 1. See [65], [79] for
example.
A basic result is that x e Cut(p) if and only if either xis the first conjugate
point to p along some geodesic from p, or there exist at least two minimizing
geodesics from p to x. For example when M = sn and p and x are antipodal then
both possibilities hold.
If r: M-+ IRis given by r(x) = d(x,p) then r is C00 on M - (Cut(p) u p)) since
there
r(x) = lexpp -1(x)lp (253)
B. Suppose now that Dis a domain in M- Cut(p) with D c W for W open with w compact and in M-Cut(p). We can use expp -1 to identify M-Cut(p) with the star
shaped open set uo of T pM, and give uo the induced Riemannian metric. Then D
and Ware considered as sets in T pM· Using spherical polar coordinates in T pM
the Riemannian metric at a point v has the form
ds2 = dr2 + ~ g· ·(v)dai dai iJ=1 1)
(254)
where a1, ••• ,an-1 refer to coordinates on the sphere sn-1. Since the space of
Riemannian metrics on any manifold (and on sn-1 in particular) is a convex set
in a linear space, it is easy to first modify gij outside of W, if necessary so
that it extends to a metric on the whole of sn-1 for each sphere in T pM about p
which intersects W, and then modify this family of metrics (one for each
relevant radius lvl) outside of D and extend so that we obtain a Riemannian
metric on the whole ofT pM of the form
218
ds2 = dr2 + ~ h· ·(v)dcsi dcsi i,j=1 l)
382
(255)
which agrees with the original one on D and agrees with the standard Euclidean
one coming from<, >ponT pM outside of some compact set.
This gives T pM a Riemannian structure for which it is complete since the
geodesics from p are easily seen to be the straight lines from p, by the
distance minimizing characterization of geodesics, and the existence of all
geodesics from some point for all time is known to be equivalent to metric
completeness. The point p is now a pole and the curvature tensors are all C00
with compact support. Moreover the heat kernel for the Dirichlet problem in D
is unchanged since all these modifications took place outside of D. We can
therefore assume that M wasT pM with this metric.
c. Assuming the metric and manifold M has been changed in this way, and M
identified with T pM,
PtD(x,p) =lim J (2n)yn/2 PtD(x,y)exp- d(y,p)2/(2:A)ep(y)dy (256) A.Ul M
where dy refers to the Lebesgue measure of T pM, identified with M, using
< , >p• and ep is the volume element from the Riemannian metric (255): in
terms of our original metric it is given on T pM by
ep(v) = ldetM Tv exppl
and is known sometimes as Ruse's invariant. See [16] for more details about
it.
Thus
PtD(x,p) = lim P to f :>-. (x) A..l-0
where P1D : t ~ 0 is the Dirichlet semigroup for ~6 + V and
f). (x) = (2n:A)-nl2 exp -r(x)2/(2:A).
(257)
To evaluate PtD f). we will use the Girsanov theorem. Fix T > 0 and for :A ~ 0
let zAt : 0 :S t < T + :A be a Brownian motion on M from a point x0 of D with
time dependent drift z:As for z:As = VYAs with
y:As(X) =- r(x)2/(2(:A+ T -s))- ~ log 9p(X) 0 :S s < T +).
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Now iff: M-+ IRis smooth with f(x) = F(r(x)) for some smooth F: (O,oo)-+ IR,
by equation (4) for 6 in local coordinates, we have
M(x) = a2F/or2 + (<n-1)/(r(x)) + o/or (log ep)(x))oF/or (258)
In particular
6r = (n-1)/r + o/or log ep. (259)
By Ito's formula for r:At = r(zAt) we have
t t t
r:At = r(x0 ) + j dr(u:As dBs) + j o/or y:As(zAs)ds + ~ j 6r(zAs)ds 0 0 0
t t t
= r(x0 ) + j dr(u:As dBs)- J r:As/(:A+ T -s)ds + ~ J (n-1)/(rAs)ds (260) 0 0 0
where (u:As ; o :s s < T + :A) is the horizontal lift of (zAs : 0 :s s < t to the
frame bundle OM and (Bs: 0 :s s < oo) is a Brownian motion (which can be taken
to be independent of :A by taking the canonical construction of z:As from an
S.D.E. on OM).
Since ldrl = 1 and u:As(w) is an orthonormal frame the martingale term in
(260) is just a 1-dimensional Brownian motion, so r:A t : o :s t < T + :A
satisfies a stochastic differential equation which is essentially independent of
the manifold M. In fact from Ito's formula it satisfies essentially the same
equation (to be precise it is a weak solution of the same equation) as for the
radial distance (lxA s I: 0 :s s < T + :A where x:A s : 0 :s s :s T + :A is the
Euclidean Brownian bridge from x0 to 0 in IRn::::: T pM in timeT +:A, given by
s
x:As = x0 - sx 0 /(T +A)+ (s-T -:A) j d~tl(t-T -:A) (261) 0
where J3s: 0 :s s < oo) is Brownian motion on IR, e.g. see [59]. This Brownian
bridge is itself equal in law to
s-+ x0 + Bs- s (BT+:A + x0 ) (T+:A)-1 (262)
Thus r:As : 0 :s s < T + :A) is equal in law to lx:Asl: 0 :s s :s T + :A). In
particular it is non-explosive: as s t T +:A so it converges top (now identified
with the origin).
220384
Lemma 1C As A .1. 0 so (zA 5 : 0 s s s T converges in law to the process
zs : 0 s s s t which is sample continuous, agrees with z0 s for 0 s s < t
and has zT = p. Furthermore zs : 0 s s s T) has radial component r(zs) : 0
s s s t which has the same distributions as the radial component of the
Euclidean Brownian bridge in IRn starting from a point distance r(x 0 ) from 0
and ending at Din timeT.
Proof Let dxt = X(xt) o dBt + A(x 1)dt be a smooth stochastic differential
equation on M whose solutions are Brownian motions on M and, identifying M
with T pM and so with IRn, such that A has compact support and X(x) = o outside
some compact set. Then we can represent zA1 : 0 s t < T + A as the solution
to
(263)
Fix t 0 E (O,T). Then zAt: 0 s t s t 0 ) converges uniformly in probability
to z 0 t: 0 s t s t 0 . (Indeed we can choose versions so that it converges
almost surely since the coefficients of (260) have derivatives bounded
uniformly on [O,t 0 ].) Now suppose 0 < t < 1 and o > 0. Set t 1 = min t/3,
to/18 and choose o 1 > o such that
IP(r0 s < t1 for t-o 1 s ssT> 1-t1 > 1- t/3,
(which is possible by the continuity of r 0 s at s = T).
Take o2 > 0 such that for o < A < o2
IPIzA5 - zsl so for 0 s ssT -o1 2:: 1-£/3
and such that
IE[sup(rAs A 1: T -o 1 s ssT] s £1 + IE[supr0 s" 1: T -o 1 s ssT]
(which is possible because rA 5 -+ rOsin probability uniformly on [O,T]).
Then, for 0 < A < o/2
IE[suprAs" 1: T -o 1 s s s T] s 2£1 + (1-£1)£1 s 3£1
whence
IP[suprAs" 1: T -o 1 :s ssT> o/2) s 6£ 1/o s £/3
and so
IP(IzAs - Zs Is o for 0 s ssT 2:: 1-£/3- £/3- t/3 = 1-£.
This prove uniform convergence of zAs : 0 s s s T to zs : 0 s s s T) in
221385
probability.
If x0 = p, we should have been a bit more careful because of the
singularity of the distance function at p, but this is no problem si nee it is only
the convergence for t near T that causes any difficulties. I I
The process zt: 0 :$ t :$ T) will be called the semi-classical bridge from
x0 top in timeT. In [42] it was called the Brownian-Riemannian bridge but
this suggestion by K.D. Watling seems preferable. In particular it emphasizes
the fact that it will not in general coincide with Brownian motion from x0
conditioned to arrive at p at time T .
D. The following comes from [81], following earlier results with more
restrictive conditions in [42], [43].
Theorem 1D Suppose 1M is a Riemannian manifold which has a pole p in the
sense that its exponential map expp maps an open star-shaped region of T pM
diffeomorphically onto IM. Let M be some open subset of 1M (possible M = IM)
with p e M. Then for V : M -+ IR bounded above and coRtinuous, the
fundamental solution to the minimal semigroup for ~6 + V on M is given by
Pt(Xo,p) = (2nt)-nl2 ep(Xo)-~ e -d(Xo,p)21(2t)
t
IE[Xt<'t)exp J (~ 9p~(zs)6ep -~(z0 ) + V(Zs))ds)] (264) 0
where zs : 0 :$ s :$ t) is the semi-classical bridge in M from x0 top in
time t, defined up to its explosion time 't. In particular the expectation on
the right hand side of (264) is finite.
Remark: By the 'semi-classical bridge' here we mean a process which is a
Brownian motion with drift Z0 s: 0 :$ s :$ t), zo as before, in the interval 0 :$ s
< t A 't) where 't is its explosion time in M (so that if 't < tit either goes out to
infinity or leaves M as s 1' 't), and which is sample continuous with value p at
timet if t < 't.
. 00 Proof: Choose a nested sequence of domains D1)i=t with smooth boundaries,
such that Di is compact, both p and x0 lie in D1
, and M is the union of the Di. Let
pit(x0 ,p) be the Dirichlet fundamental solution for of lot= ~Mt + Vft in Di.
Let 't\ be the explosion time from D; of the Brownian motion with drift
222386
ZAs : 0 :S s < T + A) starting from x. While we obtain an expression for
pit(x 0 ,p) we can assume 1M modified outside of Di as in §18. Thus "t\ is the
first exit time from o1 of (zAs: 0 :S s :S T +A).
By (257) and the Gi rsanov theorem
T
piT(x 0 ,p) = lim IE[XT<"tA·) MAT fA(zAT) exp J V(zAs)ds] (265) A..J..O - 1 o
where
T
MAT= exp (J -(ZAs(zAs), uAs o dBs> 0
T
- ~ J (-div zAs(zAs) + IZAs(zAs!2)ds) 0
for (uAs : 0 :S s < T + A) the horizontal lift of (zAs: 0 :S s < T + A in OM from
some frame u0 at x0 ; see equation (212).
Since zAs = VYAs• writing e for ep:
MAT= exp (-YAT(zAT) + yAo(xo)
T
+ J (o/os (ZAs))(zAs) + ~ IZAs(zAs)l2 + ~ 6YAs(zAs))ds) 0
T
= e(zAT)~ e(x0 )-~ exp(~ J e~ (zAs) 6e-~ (zAs) ds 0
T
- ~ r(x 0 )2/(A+T) + r(yAT)2/(2A)- n/2 J ds/(A+T-s))ds) 0
since 1 _1 1 2
-6 ~ log e(x) = e2 (x)6e 2 (x) - IV log e2 (x)!
and -6r2 =- 2- 2(n-1) -2r o/or loge by (258).
223387
T
exp -i r(x0 )2/(:A+ T)) lEX T <"tA. exp I V eff (zA s)ds (266) 1 0
where 1 1
V eff(X) = V(x) + ~ 92 (x)~ e-2 (x) (267)
To treat this limit carefully let C be the space of continuous paths o: [O,T]
-+ IM+, where IM+ is the one point compactification of IM, with o(O) = x0 • Lets :
C-+ lR u oo be the first exit time from Di. Let JPA, IP be probabilities induced
on c by zAs : 0 ~ s ~ T) and zs : 0 ~ s ~ T. By Lemma 1C, JPA-+ IP narrowly
(='weakly'). Since Di is compact and has smooth boundary
T
o-+ XT <S (o) exp I Veff (O(s))ds 0
is bounded and Riemann integrable for IP in the sense that it is continuous
except on a set of IP-measure zero (this is because there is probabHity zero of
the path of a non-degenerate diffusion hitting oDi without leaving Di). It
follows, e.g. see [88] p. 375, that
T
p iT(x0 ,p) = e(x0 )-i(2nT)-n/2 exp -r(x0 )2/(2T) IEXT <"tiexp I Veff(Zs)ds 0
(268)
If we now let i -+ oo the left hand side of (268) converges and the term
under the expectation on the right hand side is positive and non-decreasing in i.
The theorem follows. II
Remark 10. The above proof shows that the upper bound on V was not essential
provided we know that Pt(x 0 ,p) exists and is given as the limit of the Dirichlet
heat kernels.
Corollary 10. (i) [ 47] For a complete manifold M, if x0 ~ Cut(p), then
t
p1(x 0,p)?:(2nt)-n/2 ep(x
0)-i e-d(xo,p)
2/(2 t)IEXt<"t exp Iv eff(Zs)ds
0
(269)
224388
where Veff is in (267) and "t is now the first exit time of the semi
classical bridge from M- Cut(p).
Proof: Replace both ll'v1 and M in the theorem by M-Cut(p) respectively and
observe that Pt(x0 ,p) is greater than the corresponding value of the kernel for
M-Cut(p). //
Corollary 1D was used to get results about the limiting behaviour as
t J. 0 of the trace, f M Pt(x,x)dx, in [47]. It is especially useful for small t
since in the limit it becomes an equality as in Corollary 1D(iii) below. When
Cut(p) is codimension 2 it has capacity zero (the Brownian motion never hits it,
e.g. see [51]) and so fundamental solutions at (x0 ,p) for M and M-Cut(p) are the
same if x0 ~ Cut(p). Thus:
Corollaru 1D(jil c.f. [81]. If M is complete and Cut(p) has codimension 2
(or capacity zero more generally) then if x0 ~ Cut(p) there is equality in
(269). //
The following is a well known result with both analytical and
probabilistic proofs e.g. see [7], [17], 1[59a, [80], [84].
Corollary 10 (jjf) Suppose M is complete and x0 ~ Cut(p). Then as
t J. 0
Pt(x0
,p) = (2nt)-n/2 e -d(xo,p)2/(2t) ap(x 0 )-~(1 + o(t)) (270)
proof: First choose a compact domain D with smooth boundary in M which
contains the geodesic of shortest length from x0 to p. We need now quote the
result that as t J. 0
Pt(X0 ,p) = p0 t(x 0 ,p)(1 + O(tk)) (271)
for k = 1,2,... . For this see [7], [80], or [33] when x0 = p, (the O(tk) can be
replaced by O(exp(-&/t)) for some & > 0). Thus we need only examine the
behaviour of PDt(x 0 ,p) as t J. o. Choosing D with D inside M-Cut(p) we can
therefore modify M outside of D as in §18, so that p is a pole and it is flat
outside of a compact set, and also we modify V outside D to give it compact
support. Following this we can use (271) in the reverse direction and consider
Pt(x0 ,p) for the modified M. For this we have
t 1 2 ( ) f V eff<zs)ds
Pt<xo,p) = (2nt)-n/2 ap(Xo)-:z e-d(xo,p) I 2t IE e o
and so the result follows since Veff is now bounded on M. II
225389
The following corollary was noted in [42], [43]. It can be compared with
the trace formula and asymptotics in [28], [32], [94) for example. When M is
complete and p has no conjugate points then expp: T pM-+ M is a covering map
and if T pM is given the induced metric to make it a Riemannian manifold, M0 ,
say, the origin 0 is a pole. If x0 E M there are at most countably many points
x'¥ 0 in M0 with expp(x'¥ 0 ) = x0 , one for each geodesic "t from x0 top: we take 6
to be the geodesic t-+ expp tx6 0 in the reverse direction. For fixed t there is a
semi-classical bridge in M0 from each x60 to 0 in timet. Let z6s: 0 s sst
be its image in M under expp· This will be called the "semi-classical bridge
from x0 top along the geodesic 6· in timet". There is also a corresponding
e"tp(x0 ) which is just e0 (x"t 0 ) evaluated in M0 •
Corollaru 1DC1yl. Suppose M is complete and the point p has no conjugate
points. Then
1 ft (V (z6s) + cx:6s)ds Pt(x
0,p) = (2nt)-n/2 :r e"t (x )-2 e-2(6)2/21 lEe o (272)
"t p 0
where the sum is over all geodesics 6 from x0 to p,with 1(6) the length of
6· and z"ts: 0 s sst the corresponding semi-classical bridge; also cx:6s: 1 1
o s sst is ~e 0 (x6s)2 t.e 0 -2(x6s) where e 0 is Ruse's invariant in TpM
from 0 computed using its induced metric and x6 s is the semi-classical
bridge in T pM from xl 0 top in timet.
Proof. Let U be a sufficiently small open neighbourhood of x0 so that its
inverse image under expp consists of open neighbourhoods u6 of xl 0 in T pM·
Let f be the characteristic function of U and f6 that of Ul. Then for the
semigroups Pt : t 2:: 0 and Pt: t 2:: 0 for ~t. + V and ~t. + V o expp on M and
T pM respectively we see
Ptf(p) = :r6
Ptfl(p) (273)
by the Feynman-Kac formula since Brownian motion on T pM from 0 covers
Brownian motion from p in M. Because ul is mapped isometrically to U it has
the same volume as U and so we can let U be a ball radius £ about x0 and let£ J.
0 to obtain
Pt(P,Xo) = :r 1
Pt(p,x l o)
226390
in the obvious notation. However Pt(p,x0 ) = Pt(x0 ,p) and similarly for Pt(p,x'¥0 ),
and so the corollary follows from the theorem. I I
Note: The Cartan-Hadamard theorem assures us that the hypotheses on p and M
in Corollary 1D(iv) are always true when M is a complete manifold with all
sectional curvatures non-positive.
Example 10 (i) [92], [93]. The simplest non-trivial example of a manifold
with a pole is n-dimensional hyperbolic space Hn. From §4G of Chapter III,
equation (181), we see that
ep(x0 ) =(sinh r (x 0 )1(r(x 0 )))n-1 (274)
from which, using (258), we have
When n = 3 and V = 0 we can deduce the well known formula for the heat kernel
of H3:
Pt(x,y) = (2nt)-312 e-tl2 e-d(x,y)21(2t) d(x,y)l(sinhd(x,y)) (276)
with corresponding exact formulae for non-simply connected 3-manifolds of
constant negative curvature obained by using Corollary 1D(iv). The heat kernel
for the hyperbolic plane H2 is computed analytically in [28]. For a recurrence
relation between the kernels for hyperbolic spaces of different dimensions
see [28), with [35) for more details.
Example 10(111) [81], [83], [49]. ForM = sn-1 note that if p is the North pole,
say, in polar coordinates (r,<') in IRn-1 (so <l e sn-2) the exponential map is
essentially the map (r,<l)-+ (cos r, (sin r,<l)) e IR x IRn-1. In particular it maps
the sphere about 0 radius r to an embedding in IRn onto an isometric copy of the
sphere in IRn-2 radius sin r. Thus the metric in normal polar coordinates is
ds2 = dr2 + (sinr) (standard metric of sn-2).
Thus
ep(x0 ) = (sin r(x0 )1(r(x0 ) )n-1 (277)
and
1 1 1 1 ~e2(x0 ) t.e-2(x0 ) = 8 (n-1)2 + 8 (n-1)(n-3)(11r2- 11sin2r) (278)
for r = r(x0 ).
227391
Again n = 3 is an especially nice case. Since Cut(p) has co-dimension 2 we
can use Corollary 1D(ii) to get, for V = 0, if xis not antipodal toy
Pt(x,y) = (2nt)-n/2 (r/sin r) e!t e-r2/(2t) IP t < "t
where r = d(x,y) and where "t is the first hitting time of Cut(y) by the semi
classical bridge from x to y in time t. However as we saw in Lemma 1C the
radial distributions of this bridge are the same as those of a Brownian bridge
Cl3s : 0 :s s :s t, say, in IR3 from a point distance r from 0, to 0, in time t.
Thus, [81],
Pt(x,y) = (2nt)-n/2 (r/sin r) e~t e-r2
12t IPsupO:ss:so'J3sl :s n (279)
This formula is discussed in [83]. In [43] it is used to obtain the exact
formula, for x,y not antipodal,
Pt(x,y) = (2nt)-3/2 e!t L .e(6)/(sin.e('¥)) e-.e('¥)2/(2t) "t
(280)
where the sum is over all geodesics 1 from x toy and .e('¥) is the length of 6·
Note the similarity here with the case of s1, or the s1tuation in Corollary
1D(iv). However in this case we no longer have a sum of positive terms. This
formula is a special case of a general formula for compact Lie groups, [50],
proved using harmonic analysis on such groups.
§2. General remarks about the elementary formula method and its
extens1 ons.
A. The way we were able to get a tractable formula for the heat kernel in the
last section depended on a suitable choice of drift Z:>'s: 0 :s s < T +A for
which there were convenient cancellations after the use of the Girsanov
theorem, and which gave processes with a very nice radial behaviour. In fact
the choice of ZAs came from a general philosophy outlined in [42], which is
explained below. However first it should be noted that there are various ways
of getting 'bridges' from x0 to p e.g. see [ 17]. The standard one is Brownian
motion from x0 conditioned to be at p at time T. This can be described as the
h-transform of (space time) Brownian motion where hs(x) = p0 T -s(p,x) for
p0 t(x,y) the fundamental solution for M when V= 0. This is used in [36], [80]
and [ 104]; it is Brownian motion with drift Vlog PT -s<P.- ). Writing it as xs :
228392
0 s ssT it is immediate from the Feynman-Kac formula that the kernel with
a potential Vis given by
T
PT(x0 ,p) = p0 T(x0 ,p) IE exp J V(xs)ds. 0
It has the advantage over the semi-classical bridge of symmetry in x0 , p i.e.
the reversed time bridge is the bridge from p to x0 • However the radial
behaviour will not be so pleasant in general.
B. Let Pt : t ~ 0) be the heat semigroup associated to ~6 + V on M. Suppose
g0 (x) = exp(- S0 (x)) ·f 0 (x) where S0 also are smooth functions on M with S0
bounded below and with f 0 of compact support. The drift terms for the semi
classical bridge arose, [42], from seeking a nice expression for Pt g0 which
would ex hi bit its behaviour as A .1. o when S0 = A -1 R0 some R0 • Here is a
brief description. Assume for simplicity that M is complete and V, VS 0 and
the curvature tensor are all bounded on M.
First we associate to g0 the classical mechanical system with trajectories
g?t(a): t ~ O) for each a in M, satisfying
D/ot <i>t(a) = o (281)
with g? 0 (a) =a and <i>0 (a) = VS 0 (a). Under our assumptions it is shown in [42a]
that there exists T > 0 such that g? 1(a); t ~ 0 is defined for all 0 s t s T and
determines a diffeomorphism
g?t : M -+ M.
This is a 'no caustics assumption.
For this T we can define the Hamiltonian-Jacobi principle function.
s: [O,T] X M-+ IR
given by
I
S(t,a) = s0 (g?t-1(a)) + ~ J id>s o g?t-1(a)l2 ds (282) 0
There is then the following standard lemma, as in [42a]:
Lemma 28
(i) <i>t(a) = VS(g?t(a),t) OstsT, aEM (283)
(ii) S satisfies the Hamilton-Jacobi equation
229
~IVS(x,t)12 + as;at (x,t) = o with S(x,O) = S0 (x)
(iii) Define cp: N x [O,T]-+ IR by
cp(x,t) = ldet T x ~- 1 tl
393
O:St:ST (284)
(using the Riemannian metric of M). Then cp satisfies the continuity equation
acp;at (x,t) + div(cp(x,t) VS(x,t)) = o (285)
Proof:
t
VS(x,t) = (T~-1 1)* VS0(~-1 1 (x)) + J D/os T(~s o ~ 1 -1)* (<i>s o ~-1 1(x)ds. 0
Integrate by parts to obtain VS(x,t) = <i>t o ~t-1(x)
yielding (i). Also
t
a ;at S(x, t) = dS 0 ( ~-1t(X)) +~ l<i>to~t -1(x)l2+ J<D ;as T ~ s<<i>t -1(x) ), <i>so~t -1(x))ds. 0
Integrate by parts again and use (i) together with the identity
T~t o ¢-1t + <i>t o ~t-1 = 0 (286)
(which comes from differentiating ~to ~ 1 -1 = 1d) to obtain (ii).
For (iii) take any C00 function f: M-+ IR with compact support. Integrating
by parts
JM div(cp(-,t)VS(-,t))(x)f(x)dx
=- f M cp(x,t)(VS(x,t), Vf(x))dx
= - f M (VS(~t(X),t),Vf(~t(X))dx
=- f M df(<i>t(X))dx (by (i))
=- d/dt f M f(~t(X))dx = -d/dt f M cp(x,t)f(x)dx =-f M CJ/)t cp(x,t)f(x)dx
giving (iii). //
Now run the classical mechanical flow backwards. Take tin (O,T] and set
es(a) = ~t-s<~-1t(a)) 0 :S s :S t, a eM.
Then
a;as es(a) =- VSt-s<es(a)).
Let CYt: t ~ 0) be a Brownian motion on M from x0 with time dependent drift
(VY sCx): 0 :S s :S t, x eM) for Y s<x) = - S(a,t-s). We can think of it as (8s(x 0 ):
230
0 !0 s !0 t perturbed by white noise.
Formula A c.f. [42] For 0 !0 t !0 T
t
394
Pt g0 (x 0 ) = exp (- S(x 0 ,t) lE [exp(J V(ys) - ~ l::.St-s<Ys)ds f o<Yt)] (287) 0
Proof:
t
Ptg 0 (x 0 ) = lE exp J V(xs)ds- S0 (Xt)f 0 (xt). 0
Apply Girsanov's formula to obtain an expectation with respect to Ys: 0 !0 s
s t. The exponential martingale which comes in is
t
expY 0 (x0 )- Yt<Yt) + J (o/os Ys<Ys) + ~ IVYs<Ys)l2 + ~ t:.Y 5 (Ys))ds 0
t
i.e. exp S0 (yt)- S(x0,t)- ~ J t:.S(ys,t-s)ds 0
using the Hamilton-Jacobi equation. I I
This method can be modified in various ways. To obtain information about
the limiting behaviour of pXt gX 0 as X J. 0 where gX 0 is as g0 but with S0
replaced by X -1 S0 and where pXt refers to the semi group generated by ~Xt:. +
XV one proceeds in essentially the same way and Formula A gives the 'W.K.B'
approximation. However in this case, and for us, a slight modification gives a
more useful formula, [ 42a]:
Formula B
t
P tg0 (x0 )= .f <9t(x0 )exp -S(x 0 , t)lE[exp~ J<p(Zs, t-s )-~ t:.<p~ (zs, t-s )ds f 0 (zt)J (288) 0
where zs: 0 !0 s !0 t is Brownian motion on M from x0 with drift
(VY s(X) ; 0 :5 s :5 t, X E M) for
Y s(a) = - S(a,t-s) + ~ log <p(a,t-s),
assuming this process is complete.
Proof. From the continuity equation
231395
6S(x,t) = - olot log cp (x,t) - <V log cp(x,t), VS(x,t)>
so that
6S(®s(a),t-s) = oloS log !p (®s(a), t-s).
If we use this as well as the Hamilton-Jacobi equation after the Girsanov
transformation the formula follows. I I
The reason for introducing these formulae (which have many variations)
here is that to obtain the elementary formula for the heat kernel we needed the
case S0 (x) = d(x,p)21(2X). For this, given that p is a pole, we have ~ 1 (x) =
(X+ t)X -1x in normal coordinates about p and St(X) = ~ d(x,p)21(X + t). Then
'Pt(x) = (X/(X+t))n ep(x)-1 ep(XI(X+t)x)
in normal coordinates. To obtain the 'elementary formula' we could have used
the process zs = zXs of Formula Bas in [42]. However the actual process zXs
we used is easier to handle and gives the same limiting process, the semi
classical bridge, as X .1. 0.
C. This very simple approach to the study of asymptotic behaviour seems to
have wide applicability, applying to both the Schrodinger and the heat
equations. In the former there is no Girsanov theorem, but this is made up for
by unitarity of the semigroup, and the use of a transformation of semigroups:
essentially an h-transform. This semigroup approach was worked out by
Watling [9Ba] to deal with both types of equation almost simultaneously. He
showed how it could be used to obtain full asymptotic expansions with exact
remainders. This was extended by Ndumu [82], and here we give a brief
description of how to get the asymptotics of the heat kernel p1(x 0 ,p) for ~ 6 +
v. Assume that p is a pole forM, with M complete and Euclidean outside some
compact region for simplicity, and that Vis bounded and smooth. Consider
qt(x,p) = (2TTt)-nl2 ep(x)-~ exp (- d(x,p)2/(2t)) (289)
The first observation is that as a function of x, writing e for ep it satisfies 1 1
(o/ot)f t<x) = ~ 6 f t (x) - ~ e:z (x) 6e-2 (x) f tCx) (291)
and moreover as t .1. 0 it converges .to the Dirac delta function at p. Next define
the 'semi-classical' evolution (Qp(t,s) : t 2: s > 0 on bounded measurable
functions by
Op(t,s)(f)(x) = q1(x,p)-1 Pt-s<qs(-,p)f)(x) (292)
232396
where Pt: t ~ 0 is the semigroup for ~ 6 + V. Another (this time standard)
computation yields for f smooth and with compact support
(o/ot) Op( t,s )( f)(x) = 0 6 + V log qt( -,p) + V eff )Oy( t,s )( f)(x) 1 1
where Veff(x) = V(x) + ~ e:z (x) 6e-:z (x) as usual. Consequently, now by a
Feynman-Kac formula rather than a Girsanov theorem, fort> s ~ 0
t
Op( t, t-s) ( f)(x 0 ) = IE[ exp d V eff(Zr )dr f(zs)] 0
(293)
where zs : 0 :::; s :::; t is the semi-classical bridge from x0 to p in time t. Letting s i t we obtain another proof of the 'elementary formula' (264).
To get the asymptotic expansion assume now that each pair of points x
and y in M can be joined by a unique geodesic. Let l(x,y) denote this path
parametrized proportionally to arc length so that l(x,y)(O) = x and l(x,y)(1)
= y. For f : M -+ IR and r ~ s ~ 0 define
F(r,s)(f): M-+ IR
by
F(r,s)(f)(x) = f(l(p,x)(s/r)) = f(s/r x)
in normal coordinates at p. Then, as for (292), for smooth f of ·compact
support
(o/os) [Qp(t,t-s)F(t-s,t-r)(f)J(x) = Op(t,t-s)~ [F(t-s,t-r)(f)J(x) (294)
where
y = ~!1- ~Vlog9.V+Veff· From (294) we have on integrating
Op(t,t-s) [F(t-s,t-r)(f)J(x)- Op(t,t)[F(t,t-r)(f)J(x)
s
= J Op(t,t-s 1) y[F(t-s1, t-r)(f)J(x)ds 1• 0
Setting r = s we get an expression for Op(t,t-s)(f) since F(t-s,t-s) = 1d. This
can be iterated arbitrarily many times by replacing f by g[F(t-s,t-r)(f)] and
substituting in the integrand, to yield a rather complicated expansion for
Op(t,t-s)(f) to arbitrarily many terms with a remainder consisting of a time
integral of various iterations of the operators. Knowledge of Op(t,t-s)(f)(x0 )
gives knowledge of Pt(x 0 ,p), c.f. (292). This way Watling's expansion [98a],
233397
[98] is obtained, see also [82], for N = 1,2, ....
Pt(x0 ,p) = (2nt)-n12ep(x 0)-~ exp( -d(x0 ,p )2 /(2t) )[ 1 + a 1 (x 0 ,p )t
+ ... +aN (x0 ,p)tN] + RN+l(x 0 ,p,t)tN+1 (295)
where
and
1
a1(x0 ,p) = J F(1, 1-r1)(Veff)(x)dr 1 0
aj<x0 ,p) = j i' ... t1
F(1,1-ri).F(1-ri'1-rj-1l ... 0 0 0
... ~F(1-r2,1-r1)(Veff)(x)dri ... dr 1
for 2 !i j !i N and
1 rl rN
RN+l(x 0 ,p,t) =lEd J ... J (gF(1-rN+l•1-rN) ... 0 0 0
trN+l
... ~F(1-r2, 1-rl)(Veff))(ZtrN+l) exp J Veff(Zs)ds)drN+1 ..• dr1].
0
As before this gives an exact expression when M has a pole and given some
additional bounds on its geometry [82], and furnishes an asymptotic expansion
when x0 $ Cut(p) for general complete M. It is easily modified to deal with the
fundamental solution to ~ 6 + A + V where A is a first order operator (i.e. a
vector field). Essentially the only differences are: (i) that 1
ep -2(x) is replaced by
1
ep -~(x) exp ( J <t(s), A(0(s))ds) 0
where 0 is the geodesic from x to p parametrized to take unit time, and (ii) ~6
is replaced by ~6 +A throughout; see [98].
234398
§3. The fermionic calculus for d1fferential forms, and the Weitzenbocl<
formula
A. The use of creation and annihilation operations for differential forms was
exploited by Witten for his approach to Morse theory (99]. The notation is very
useful in stating, and proving, the Weitzenbock formula for the Laplacian on p
forms, as described in [33]. For our purposes it will enable us to give an
'elementary formula' for the heat kernel for forms especially suited to the
'supersymmetric' approach to the Gauss-Bonnet-Chern theorem which is
discussed later. It would be difficult to improve on the exposition in [33] and
it will be followed closely, as in [48] on which these sections are based.
B. First let us fix some notation and recall some basic facts. If Vis a real
finite dimensional vector space the space AP(V) of antisymmetric linear
maps cp: V x ••• x V-+ IR can be identified with the space of linear maps ll...(APV;
IR) i.e. (APV)*. If cp e AP(V) and ex e A 1(v) there is ex A cp e AP+ 1v given by
ex A cp(v 1, .•. ,vp+i) = !,1
(-1)i+1 ex(vj) cp(v 1, ... ,vj•···•vp+i) ]=1
(296)
where A indicates that the indicated term is omitted. This determines an
isomorphism of (APV)* with APV*, every element of the latter being
representible as a linear combination of terms of the form
ex1 A ••• A exP for exi E V*. If V has an inner product there is an induced inner
product on APV and APV* determined by
1 1 i . p (ex A ••• A cxP, J3 A ••• A J3P) = det [(ex, J3l)Ji,j=l
for exi, pi in V or V* respectively.
For such V, given cp e APV* and e e V define the "creation operator" a(e)*:
APV*-+ AP+1 V* by
a(e)*cp = e• A cp
where e• E V* is dual to e (it is most convenient to formulate it this way to
avoid a plethora of •·s later on). Let its adjoint be
a(e): AP+1y*-+ APV*.
Then
235399
a(e)(a111 ••• II aP+1) = ! 1
(-1)i+1ai(e)a111 ••. II ai II ••• II aP+1 (297) H
or as an antisymmetric linear map, for cp e AP+1(v)
a(e)cp(v 1, ••• ,vp) = cp(e,v 1, ••• ,vp) (297a)
There is the anti-commutation relation for
(a(e),a(f)*) := a(e)a(f)* + a(f)*a(e): APV*-+ APV*
withe, f e V:if cp e APV* then, from (297)
(a(e),a(f)*)cp = (e,f)cp (298)
C. A p-form cp on M gives an anti-symmetric p-linear map
fPx : T xM x ••• x T xM -+ IR
for each x e M. Thus we can consider fPx e AP T x*M and consider cp as a
section of the tensor bundle APT*M. Let AP denote the space of such sections
which are C00 and let A = ffip AP be the space of sections of A T*M := EBP APT*M. Supposing M is given a Riemannian structure (as we will from now
on) we can use the Riemannian measure and the inner products in each APT*xM
defined as above to obtain a space L2AP of l2 p-forms with inner product
(cp,\P)L 2 = f M <fPx• 'Px>dx.
Exterior differentiation d : A-+ A restricts to
d: AP-+ AP+1
for each p with
( dcp )x(v 1, ... ,vp+ 1) = f,\ -1)i+ 1ocp(x)( vi)( v 1, .•. ,v j, ... ,vp+ 1)
in local coordinates, where D is the Frechet derivative. Since the covariant
derivative agrees with the ordinary derivative at the centre of normal
coordinates for the Levi-Civita connection
(299)
From the definition we gave of covariant differentiation by lifting to OM it is
almost immediate that if cp, 'PeA then for v e TM
Vv(cp 11 ~P) = Vvcp 11 'P + cp 11 Vv~P (300)
236400
Let e1, ... ,en be an orthonormal basis forT xM then if <p e AP and v1, ... ,vp e
TxM
f..ca(e;)* Ve.<tJ)x(v 1, ... ,vp) = 2,. et AVe. <p(v 1, ... ,vp) j=l ' J j=1 J
= 2:i 2:k(-1)k+1<ej,vk> Vei'P(v 1, •.• ,vk, ... ,vp)
= 2:k(-1)k+1 v vk <p (v1, ... ,vk, ... ,vp).
Thus from (299) the two types of differentiation are related by
(d<p)x = 2:nj=1 (a(ej)* Vei ~P)x
Since a(ej)*(<p A If') = a(ej)*(<p) A If' = (-1)P <p A (a(ej)*IJ!)
equations (300) and (301) immediately yield
d(<p A I')= d<p A If'+ (-1)P !p A dl'
Consequently if cx1, .•• ,o::P are 1-forms
1 - "+1 < '+1 d(o:: " •.. " o::P)- 2:(-1)1 0::1 A ••• " do::l A o::J A ••. A o::P
(301)
when <p e AP
(302)
(303)
Let d* be the formal L 2 adjoint of d so d*: A-+ A restricting to
d*: AP+1-+ AP for each p. Thus d* = o in the notation of Chapter IV.
We already know, by (39) and (40), that the formal adjoint of V acting on
functions is minus the divergence: for a vector field A
V* A(x) = -di v A(x) = - Lj <V ejA(x),ej>·
Thus on 1-forms o::
(d*o::)x = - 2:j a(ej) V ejo::.
From (303) it follows that for <peA the same formula holds:
(d*~P)x =- Lj a(ei) V ej <p (304)
D. Let A be a section of IL(TM;TM), so for each x we have Ax:T xM-+ T xM. It
has adjoint
A* x : T x *M -+ T x *M
and can operate on A by
(A" (o::1 A •.. " o::P))x = -t o::1x" ••• "A*x(o::ix)" •.. A o::Px J=l
Observe that
A"x = -2:j,k A *kl a(ek)* a(e1)
(305)
(306)
237401
where A*kl"' <el, Aek>·
For vector fields A, B the curvature tensor R determines R(A(x), B(x)) : T xM ~
T xM for each x e M, which is skew symmetric. Recall from equation (83) that
if Vis another vector field
R(A(x), B(x))(V(x)) = ([V A• V 8 ]- V[A,B])(V)(x).
It is therefore immediate from (300) that as operators on A
[V A• V 8 ]- V[A,B] = R(A(·), B(·))" (307)
Recall that the (de-Rham-Hodge) Laplace operator 6: A-+ A is defined by
6 =- (dd* + d*d)
using the sign which makes it negative definite since d2 = 0 : see equation
(228).
Set Rijkl = R(ei,ej,ek,e 1) = <R(e;,ej)el,ek>• see (86). Note the sign difference
from [33], [88].
Proposition 30 (Weitzenbock formula)
6 = trace v2 - w (310)
where W, the Wettzenbock term, is the zero order operator given at x by
Wx = -L . . k 1
RiJ.kl a(e;)* a(el·)a(ek)* a(e1) (311) 1,), '
Proof Take normal coordinates at x, and using e1, ... ,en as a basis for T xM,
take E1, ... ,En to be vector fields on M which are C00 with compact support and
agree with the Gram-Schmidt orthonormalization of the fields a;ax1, .•. ,a;axn
at each point near x. Then for y near x, E1(y), ••• ,En(Y) forms an orthonormal
base for T yM· Moreover the covariant derivative of each ei vanishes at x.
Write a(j) for a(Ej( ·)) acting on forms by ( a(j)cp )x = a(Ej(x) )cpx, etc. By (30 1) and
(304) if cp e AP, summing over repeated suffices and working near x
dd*cp =- a(j)* vi a(k) Vkcp
so
Also
d*dcp = -a(k) V k a(j)* Vi cp
so
(d*dcp)x = -a(ek)a(ej)* (Vk Vjcp)x·
Thus
238402
(t:.cp)x = a(ei)*, a(ek) (V'Vkcp)x + a(ek)a(ei)* ([Vk,Vj]cp)x
=trace (V2cp)x + a(ek)a(ei)* R(ek,ejf
by (307) since [Ek,Ej] = 0 near x. Thus (310) holds with
W =- a(ei)a(ej)* R(ei,ejt (312)
= -a(ei)a(ej)* <R(ei,ej)ek,e1)a(ek)*a(el)
by (306), which agrees with (311) since a(ei), a(ej)*) = oij and R(ei,e;)
= o. II
Let t:.P : AP -+ AP denote the Laplacian acting on p-forms (i.e. the
restriction oft:.) and wP the corresponding Weitzenbock term.
The following special case of the Weitzenbock formula was used in
Chapter III:
Corollary 30 For a smooth 1-form cp
t:.1cp = ~ trace yr2cp- Ric(cp•,-),
Proof By (312) for veT xM
(W2cp)(v) = a(e;)a(ej)* R(e;,ej)*(cp)v
= a(ei) (ej •" (cp o R(ei,ej)(-))) (v)
= -<ej,v> cp(R(e;,ei)(e;))
= -<R(e;,v)(e;),cp•> = Ric(cp•,v). II
§4. An elementary formula for the heat k:ernel on forms
A. Assume M is complete. Then the de Rham-Hodge Laplacian on C00 forms
with compact supports is known to be essentially self-adjoint, [52], [91], [33],
and so determines a semigroup e~tt:. : t;:::: 0 on L2A which by elliptic
regularity has a kernel kt(x,y) which is C00 in t > 0, and x,y in M such that for
cp in L2A
(e~tt:. cp)x = JM kt (x,y)cpy dy (313)
with
kt(x,y): AT y*M-+ AT x*M.
By the same argument described for 1-forms before the proof of Theorem
3B of Chapter IV the Weitzenbock formula implies that if WP is bounded below
then e~tt:. determines a bounded map from L 00 Ap to L 00 AP and if cp e L2AP
239
is also bounded then
e~tt. r.p = Ptr.p
where
P1r.p(v 0 ) = IEr.p(vt)
403
for v0 E APT x0M with Vt(w) e APT Xt(w)M given by
(D/ot)(vt) = - ~ (WP )*(v1) Xt
(314)
(315)
along the paths of the Brownian motion xt: t <= O) from x0 • Here we must also
assume that M is stochastically complete (e.g. that Ric is bounded below on M).
Here (WPx)* is the dual of WPx: (APT xM)*-+ (APT xM)*.
B. To obtain a formula for the kernel like that of §1, assume that p is a pole
for M, work with q-forms (to avoid confusion!), take ex e Aq T p*M and choose
r.p e AQ with compact support and such that f.Pp =ex.
Fix "t > 0 and for A. > 0 define f.P>.. e Aq by
f.P>..,x = (2nA. "t)-n/2 exp -d(x,p)2 /(2>.. "t)r.px (316)
Observe that the l<erneli<Qt(x,y) for q-forms satisfies
I<Qt(x0 ,p )( o::)(v 0 ) = limA..l.O P tf.PA. (v0 ) (317)
In fact we will obtain a formula in a slightly different form to that in §1
and more adapted to describing the asymptotics as t J. 0. For this let Ht: t <=
O) be the semigroup et"tt./2 : t <= 0). Thus Ht = Pt"t· Let Xt : t <= 0 now have
generator ~ "tt.0 (where 6° is the Laplacian on functions) and let vt: t <= 0 be
defined by
(D/ot) (vt) =- ~ "t(Wxt)*(vt) (318)
Then
(319)
As in §1 apply the Girsanov theorem to obtain
P"t f.P>.. (v0 ) = e(zA. 1 )~ e(x 0)-~ (2TT"t(1+A.))-n/2 exp -d(x0 ,p)2 /(2(1+A.)"t)
1
IE[exp("t J ~ e~(zA.s) t. 0e-~(zA.s)ds)r.p(vA.1)] 0
(320)
where the processes (zA.s : 0 ~ s < 1 + A.) now have generators ~ "tt. 0 + vy>..s
for
240404
YAs(X) =- ~ d(x,p)2/(A+1-s)- ~'t log 9(x) (321)
and vAs: 0 :S s < 1 +A satisfies the analogue of (318) but along the paths of
z\ : 0 :S s < 1 + A.
To take the limit as A J. 0 we need to know that lim vA 1 exists, and to get a
sensible answer we would like this limit to be limsil v0 s. In particular the
latter should exist. For simplicity assume now that in normal coordinates
about p the manifold M is Euclidean outside some compact set.
Proposition 4A For each v0 e APT x M the limit limsil+A vAs exists almost 0
surely. Moreover in normal coordinates about p it exists in L2 uniformly in
0 :S A :S 1 and 0 < 't :S 1.
Proof From §4E of Chapter II and the stochastic version of equation (12), in
normal coordinates about p for 0 :S s :S t < 1 +A
t t
vAt- vAs=- J r(xAr)(odxAr)vAr- ~ 't J (Wq A )*(vAr)dr s s x r
Since WP is bounded lvArl is bounded independently of chance, and of
A, 't, r. Also !r(x)l < canst. lxl since it vanishes at the origin and has compact
support. Thus there exist constants c1,c2···· independent of s,t, A, 't and of
chance such that using the Euclidean inner product of our coordinates:
t
lvAt- vAsl :S c 11 J r(xAr)(J.l uAr o dBr)l s
t t
+ c2 J lxArl/(1 +A - r)dr + c3 J IV log e(xAr)ldr s s
(322)
where J.l2 = 't, uAr : 0 :S r <A + 1 is the horizontal lift of (xAr : 0 :S r < A+1,
and Br: 0 :S r < oo is a Brownian motion on !Rn. The radial component JxArJ is
now such that if pAr= J.l-1lxAr1 then pAr: 0 :S r :SA+ 1 has the same law as
the Euclidean Brownian bridge from J.l-1 x0 to 0 in time 1 +A, by the same
241405
argument as for the case Jl = 1. Thus it itself is equal in law to
r-+ lx0 + Jl Br- Jlr(B1+X + Jl-1x0 )/(1 +X)!
i.e. r-+ 1(1 +X- r)(1 + x)-1x0 + Jl(1+X)-1(1+X-r)B 1+X + Jl(Br-B 1+x)l
Thus the second term on the right hand side of (322) can be estimated by
t
c2 f I(1+X)-1x0 + J1(1 + X)-181+>.. + J1(1+X-r)-1(Br-B1+X)Idr s
t
~ c2 (t-s) + c3 IB1+XIIt-sl + c4 f IB1+X- Brl (1 +X- r)- 1dr s
= 0(11 + X - slex)
as s i 1 +X both almost surely, and in L2 independently of .Jl, X, for any
ex e (0,~). (using the pathwise Holder continuity of Brownian paths for the
almost sure case). The martingale term in (322) is equally tractable: since r is bounded and so is luX rl , it is a time changed Brownian motion with a
bounded time change. Thus, as s-+ 1 + X,
lvX1 - vXs! = 0(11 + X - slex)
for any ex e (O,~) both almost surely, and in L2 uniformly in Jl e (0,1] and X e
[0,1]. This gives the required result. II
Theorem 48 Let M be a complete manifold with pole p such that the
Weitzenbock term for q- forms, Wq, ts bounded below. Let (zs : 0 ~ s ~ 1
be the semi-classical bridge from x0 to p in time 1 with diffusion
constant "t: so it has ttme dependent generator ~"t6 + VY0 s, 0 ~ s < 1, for
yo given by (321). For v0 e .t\ qT x M let vt : 0 ~ t < 1) be the solution to 0
(318) along the paths of (zt : o ~ t < 1. Then
(i) v1 = limt-+oo Vt exists almost surely as an element of .t\GT pM;
(ii) the fundamental solution to the heat equation for q-forms is given by
kG"t(x0 ,p)(ex)(v0 ) = (2n"t)-n/2 ep(x 0)-~ exp( -d(x0 ,p)2/(2"t)
1
IE[exp~"t J ep(Zs)~.1ep -~(zs)ds ex(v 1)]. (323) 0
242406
Proof Part (i) follows from the previous proposition by progressively
modifying the metric of M to be Euclidean outside of larger and larger domains
as described in §18.
00 For (ii) take domains Dii=l as in the proof of Theorem 1D and observe
that k q 't ( x 0 , p) = lim k q • i 't ( x 0 , p) where k q • i 't refers to the fundamental i~oo
solution on Di with Dirichlet boundary conditions. To compute kq,i't we can
assume M is Euclidean outside of a compact set, in normal coordinates about p.
Use the canonical S.D.E. on OM with added drift to define xAs: 0 :S s < 1 +:A.
Then by standard results about S.D.E. with parameters as in [57] or [90] there
are versions of x:As : o :S s < t) and vAs : o :S s :S t for each t < 1 such that
v:As converges in L2 (in our coordinates) to Vs uniformly in s E [O,t]. By the
previous proposition it follows that v:A 1 ~ v1 in L 2 and so arguing as in the
proof of Theorem 1D
kq,i't(x 0 ,p) = (2rn)-nl2 e(x 0 )-~ exp -d(x 0 ,p)2/(2't)
1
IE[X; exp ~ 't J ehzs) 68-~(zs)ds cx(v 1)] 0
where Xi is the characteristic function of w E Q: Zs E Di for 0 :S s :S 1). Now
take the limit as i ~ oo: the result follows by dominated convergence since v 1
e L 00 and the exponential term is in L 1 by Theorem 1D. II
Example 48 (Hyperbolic space). For M = Hn the Weitzenbock term Wqx is just
multiplication by the constant -q(n-q). It follows that in (323)
V1 = exp nq(n-q)'t //V 0
where //v0 refers to the parallel translate of v0 along the paths of Zt: o :S t
:S 1. Thus for ex E A qT*pM and v0 E A qT x0
M
kq't(x0
,p)cx(v0
) = (2rn)-n/2 (r 1 si nhr )~ (n-1) e-r 2 /(2't )e~ q(n-q)'t
(324)
1 1 with 82 Czs)6e-2(zs) given by (275), and in particular equal to -1 when n=3.
243407
CHAPTER VI. THE GAUSS-BONNET -CHERN THEOREM
§1 Supertraces and the heat flow for forms
A. Let V be a real inner product space. If 8 e 0...(/\V*; /\V*), with gq its
restriction to /\qV*, define its supertrace, str B, by
str 8 = Lq (-l)q trace Bq.
For a fixed orthonormal basis e1, ... ,en of V we have the annihilation and
creation operators a( ei) and a( ei )* defined in §3 of the previous chapter. For I
= i l>····ik a naturally ordered subset of l, •.. ,n define
ai = a(ei1
) ••• a(eik) e 0...(/\V*; /\V*).
Followers of quantum probability will recognise these and recall, e.g. from
[78a] p. 221, that the collection (ai)* aJr,J forms a basis for D...(/\ V*;/\ V*).
Indeed there are the correct number of them, 22n, and they act transitively on
/\ V* since given ex e /\ V*, with ex * 0, it can be annihilated down to a non-zero
element of /\ 0 V* ~ R by a suitable aJ and any non-zero element of /\V* can be
created from a non-zero element of /\OV* by a suitable (al)*. Al1ernative1y
see [33] page 248.
A basic result, emphasized, and called the Berezin-Patodi formula in [33]
is:
Propositlon 1A. ForB= LI,J f3r,J<a1)* aJ,
str B = (-1)n .J31, ..• ,n), l, .•• ,n (325)
B. The de Rham cohomology groups, or vector spaces, Hq(M,IR) were defined
by (243) in §4A of Chapter IV. As described there, when M is compact, the
Hodge theorem shows that
dim Hq(M;IR) =dim ker t.q.
(The latter is finite by ellipticity of t.q). The Euler characteristic for
compact M can be defined by
X(M) = Lq (-1)q dim Hq(M;IR) = Lq(-1)q dim ker t.q.
Compactness of M implies that if Ptq = exp ~ tt.q acting on L2Aq then P1q
244408
is trace class fort> 0 and
trace Ptq = J M trace ktq (x,x) dx
e.g. see [33]. Define the supertrace of Pt = EBq Ptq acting on L2A by
str P1 = Lq (-l)q P1q t > o. There is the following remarkable fact due to McKean and Singer:
Proposit1on 18 ForM compact and all t > o StrPt = X(M). (326)
Proof Divide A into A+ EB A- where A+ = EBp even AP and A- = EBp odd
AP
respectively and for A e IR let n+(A) and n_(A) be the multiplicities of A as an
eigenvalue of 6+ and 6-, (the spectrum of 6 consists of a discrete set of
eigenvalues increasing to oo, see [28] for example).
Note that 6 = -(d + d*)2 since d2 = 0. Therefore if 6rp = AfP then 6(d+d*)rp
= A(d+d*)rp. Also (d+d*)rp * 0 unless drp = o and d*rp = 0 i.e. unless 6rp = 0.
Thus n+(A) = n_(A) for A * 0. However
Str Pt =LA (n+ (A)- n_(A))e-~tA Therefore
Str Pt = n+(O)- n_(O) = X(M)
as required. I I
From the proposition there is the following corollary
Corollaru 181) For all t > 0, when M is compact
X(M) = I M str kt(x,x)dx
§2. Proof of the Gauss-8onnet-Chern Theorem
(327)1 I
A. Suppose M is compact with even dimension, dim M = 2.g, say. The
G-8-C. theorem expresses X(M) as an integral over M of a certain function of
the curvature of M. We can do this by looking at (327) as t J. 0 using the results
of §1 and the 'elementary formula' (323) for the heat kernel. This is
essentially Patodi's proof as described in [33]. The difference from the
treatment in [33] is simply the use of the elementary formula. The Malliavin
calculus approach as in [59a] has the same structure but the cancellations take
place at the level of distributions on Wiener space i.e. before taking
245409
expectations. Related proofs of other classical index theorems are in [18],
[56], [59a], [68], [103], [104], and discussed ln [33]. The approach of [104]
fits in particularly well here, especially if it is simplified somewhat by using
semi-classical bridges: an important technique used in [104] is a rescaling of
Brownian bridges, and this seems to work equally well with semi-classical
bridges. The Atiyah-Singer index theorem describes the index of an elliptic
differential operator ~ (i.e. dim ker ~ - dim ker ~*) in terms of the
coefficients of ~. For the G-B-C. theorem ~ = d + d *:A+-+ A-. (Remember
t.rp = 0 iff d rp = d*rp = 0). The other 'classical index theorems' are for other
geometrically defined operators~. It turns out that the general result, for~ of
arbitrary order, follows from these special cases.
B. To examine the behaviour of str k,;(x,x) as 1: J. 0, or more generally str
k,;(x 0 ,p) when x0 • Cut(p) we can argue as Corollary 1D(iii) of Chapter IV and
assume that p is a pole for M and M is Euclidean outside some compact set in
normal coordinates about p. (Of course we have lost the compactness of M
after this modi fi cation.) Having done this, rewrite (323) as
kq,; (x 0 ,p) = (2n,;)-n/2 e(x0)-~ exp -d(x0 ,p )2/(2"t)
1
IE[exp !"t J Veff(Z("t) 5)ds~q"t] 0
where
~ q"t : A qT;M -+ A qT:0M
is given by
~q,;(cx)(v0 ) = cx(v 1).
(328)
using the notation of (323), but now we write Zs as 2(1:) 5 to make clear its
,;-dependence. Writing ~1: for ffiq ~q"t note that ~1: = ~1:,1 where ~,;,s:
0:$ s :$ 1) is the solution to
(D/os) ~1:,s =- ~ 1: ~1:,s Wz(,;)s (329)
along the paths of z(,;)s : 0 :$ s :$ 1) with ~1:,0 = 1d. This exists by
Propositi on 4A of the last chapter.
Now take an orthonormal base e 1(o), ... ,en(O) forT x0
M and let ej"t(s) =I Is 1:
246410
ei(O) be the parallel translate of ei(O) along 2(1:) from x0 to z(1:)s· Omitting
the superscript 1: unless it is needed, write
Rijkl(s) = (R(ei(s), ej(s)) e1(s), ek(s))
and set ais = a(ej(s)), a random annihilation operator. If we parallel translate
~1: s back to x0 and define H1: s E IL(A T:flM; AT: M) by • • v 0
then
for
~(s) = -i: Rijkl(s)(ai)* ai(ak)*a.e
where ai = ai 0 = a(ei(O)).
C. If we iterate the formula
1
H1:, t = Id - ~ 1: J H1:,s o~(s )ds 0
(330)
by substituting the corresponding expression for H1:,s back into the integrand
we obtain
H1:,i = Id + z1 + ••• Z,e + 0(1:h1)
where
1 St 8z
Z.e = (- ~ 1:).e J J ... J ~(s,e) o ... o ~(s 1 )ds 1 ... ds.e 0 0 0
for .e = 2,3, ... , and analogously for ..e = 1.
Thus by the Berezin-Patodi formula (325)
and
1 Sl Sz
str Z.g = (- ~ 1:)..e J J ... J z1:(s 1, ... ,s.g)ds 1 ... ds.g 0 0 0
where
(331)
(332)
247411
z't(s1 , ... ,SJ) = ( -1)1 ~ sgn(1t)sgn(cr) R1t(1)cr(1)1t(2)cr(2)(st) ...
··· R1t(n-1)cr(n-1)1t(n)cr(n)Cs1) where the sum is over all permutations 1t and cr of 1, ... ,n. (This is still a random
variable.) To get Z.t into the form in which we could read off its supertrace by the
Berezin-Patodi formula we have used the anti-commutation relations ai(aj)* + (aj)* ai
= oij and ai aj ;: -aj ai.
Now 4>1 ;: (I I 1 0 )* H-r,1 ;: H-r, 1 + [(/I 10 )* - IdlH-r, 1 . We claim that
str(l 11 0 )*-IdlH-r,1 = 0(-r), see §E below. Then from (328), (331) and (332) 1
strk-r(x0 ,p);: (-41t)-lep(x0 )-2 exp -d(x0 ,p)21(2-r)
1 lSl S2
lEiexp d 't J v eff (z('t)s)ds J J ... J z't(s1·····S.t)ds.t···ds.tl + O('t) (333) 0 0 0 0
Now let 't J, 0. Still with our assumptions on M we can choose versions of z('t)s : 0 s s
s 1 so that, almost surely, they converge uniformly on [0,1] to the geodesic from x0 to
p parametrized to take unit time. (Recall from (321) that z('t) had generator ~ 'tL\ 0 +
VYs0 with
Y sO(x);: - ~ d(x,p)21(1-s)- ~ 't log O(x).)
Correspondingly by Proposition 4A of Chapter V the horizontal lifts u('t)s:Osss1 will
converge to that of the geodesic, and z-r will converge to the corresponding non
random term Z0 . In particular as 't J, 0 so str k-r(p,p)-+ E(p) where
E(p) = (41t)-.t11(l!) L sgn 1t sgn cr R1t(1)cr(1)1t(2)cr(2) (p) ...
R1t(n-1)cr(n-1)1t(n)cr(n)(p) (334)
for Rijkl(p);: <R(ej(O), ej(O))eJ(O), ek(O)>.
In particular the right hand side of (334) does not depend on the basis e1 (O), ... ,en(O) for
T pM. Note that, by equations (85) and (84 ),
R1t(l)cr(1)1t(2)cr(2) = -Rcr(1)cr(2)1t(2)1t(1)- Rcr(2)1t(1)1t(2)cr(1)
= R1t(1)1t(2)cr(1)cr(2) + R1t(1)cr(2)1t(2)cr(1) ··
-1
Thus E(p) = ~ Lsgn1tsgncr R1t(1)1t(2)cr(1)cr(2)(P) R1t(3)cr(3)1t(4)cr(4)(P) (41t) l!
... R1t(n-1)cr(n-1)1t(n)cr(n)(P)
248412
-l
= ··· = ~ L sgn1tsgncr R1t(1)1t(2)cr(1)cr(2)···RJt(n-1)1t(n)cr(n-1)cr(n) (41t) l!
which is the more standard expression for it.
D. From (327), Corollary 1B(i), we obtain:
Gauss-Bonnet-Chem Theorem. For a compact even dimensional manifold M
X(M) = f M E(x)dx where E is defined by (330).
The only thing we need to be careful about is the uniformity in x of the analogue
k-r(x,x) = k-rD(x,x) (1 + 0(-rk))
of (271) for suitable domains D which enabled us to replace M by a manifold with x as
a pole.
Special case: dim M = 2 Here
E(x) = (41t)-1 -R1221(x)- R2112 = (21t)-1 K(x)
where K is the Gauss curvature, see (87). Thus we have the classical Gauss-bonnet
theorem X(M) = (21t)-1 f M K(x)dx.
E. To see that str((/ 110 )*-Id)H't,l = O('t) consider the expansion of (I I 10 )*-Id in
terms of ai, aj* analogous to that for H-r,1 , using normal co-ordinates as in the proof
* of Proposition 4A. As an operator on AT x M , if us = u('t)s is the horizontal lift of 0
z('t)s , 0 ~ s ~ 1 , then
1 1
1
(I 110 )*-Id = J ~ (z('t)s) (odrl('t)s) aj* ak + 0
J Jr~\(z('t)sl)(odzjl('t)sl) r~z<z)('t)s2)(odzh('t)s2) .1ak1 a~2ak2 + .... 00
If we substitute dz(t)s = )lU('t)sodBs-(z('t')81(1-s) + hV log 9(z('t)s))ds,
and use the facts that r(x 0 ) = 0 and ).t -llz('t)sl, 0 ~ s ~ 1 , is a Bessel Bridge in IRn
from 0 to 0 it is easy to see that the p th term in this expansion is 0( 'tP) , p = 1 ,2, ....
Now multiply by the expansion of H't,1 and use the Berezin-Patodi formula (325).
See also [59a]. I I The expansion of (I 1 10 )*-Id plays a more important role in other index
theorems: [18], [59a], [104].
249413
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258
NOTATION INDEX
A , 400 A*, 298
ad(g), 298 a(e), a(e)*, 398 Cut(p), 380 d(-,-), 304 df, 284 dφ, 399 d*, 400 div, 307 D/t, 300 Exp, exp, 297, 301
ijg , 306 g ,ij 304
grad, 306 GLM, 297
nH , 315 H ,u 299 HTB, 298 K ,p 321
qtk (x, y), 402, 405
(TM;TM), 301 o(n), 313 O(n), 296 OM, 305
p (x, y),t 364
R(-,-), 320 Ric, 321
98 SO(n), 296 SOM, so(n), 312 TM,T M, T f,x x 283 T*M, 301 VTB, 297 α ,x 317
ijk, , 298
δ, 368 ∆, 307, 358
p, 368, 372 θ, 324
,t 331 p
Θ, 324
( ),0 364 , 335
Ω, 324 , 298
/ i,x 300, 303 <-,->x, 304
2, , 302, 306 |A|, 342
L
p,A 371
g,
, 382
M
s , 2
259
INDEX
A Action, 296–298, 305, 312, 313, 315, 342,
353, 354 Adjoint action, 296 Annihilation operator, 398, 407, 410 Atlas, 282, 283 B Bochner’s theorem, 359, 367, 368, 416 Bridge
Brownian, 383, 384, 391, 404, 409 semi-classical, 385, 388, 389, 391, 392,
395, 396, 405, 409 Brownian bridge, 383, 384, 391, 404, 409 Brownian motion
with drift, 308, 366, 385, 391 flow, 294 1-form, 311 on manifold, 280, 416l S.D.E., 308–310, 317, 383
C Cartan development, 312 Cartan-Hadamard theorem, 380, 390 Characteristic exponents, 331–359 Chart, 283, 284, 292, 295, 298, 300–302,
304, 306, 350, 380 Christoffel symbols, 298, 299, 305 Cohomology
with compact support, 371–374 de Rharn, 371
Complete manifold, 374, 378, 380, 387, 390, 405
Conjugate points, 380, 381, 389 Connection
affine, 300, 304, 308, 311, 320, 324, 327–329
form, 298, 305, 314, 323, 326, 327, 342 Continuity equation, 336, 338, 394 Cotangent bundle, 301 Covariant
derivative, 288, 299, 300, 302, 305, 306, 310, 317, 330, 339, 378, 401
equations, 335, 358, 362, 363 Covering space, Riemannian, 347, 348,
375, 380
Creation operator, 398, 407 Curvature
constant, 322, 326, 335, 347, 348, 351, 359
form, 323, 324, 326 Gaussian, 321, 322, 345 mean, 319 non-positive, 355, 380, 390 Ricci, 294, 321, 322, 345, 359, 363,
367, 369, 374, 378 sectional, 322, 374, 380, 390 tensor, 321, 325, 348, 378, 382, 392,
401 Cut locus, 380 D Diffeomorphism, 283, 284, 291–293, 296,
297, 301, 305, 312, 315, 324, 327, 331, 334, 341, 346, 350, 374, 377, 378, 380, 392
group, 291 Divergence, 307, 336, 367, 400
theorem, 307, 336, 367 Drift, 308, 362, 363, 366, 382, 385,
391–394, 406 E Elliptic, 306, 337, 341, 363, 364, 402, 409,
420 Embedding, 280, 283, 284, 286,
289–291, 293, 295, 305, 311, 317, 334, 338, 341, 352, 390
Ends, 315, 373, 375 Entropy, 337, 341 Euler characteristic, 345, 407 Explosion time, 280, 285, 287, 293, 308,
385 map, 293
Exponential map, 284, 297, 301, 316, 327, 379, 380, 385, 390
Exterior derivative, 323 F Feynman-Kac formula for forms, 280,
379, 392 Flow map, 288, 290, 293, 319
260
Frame linear, 297, 312 oriented, 312, 315, 353 orthonormal, 305, 308, 312, 313, 342,
383 Fundamental solution, 366, 379, 385, 388,
391, 397, 405 G Geodesic, 299–304, 309, 316, 327, 328,
349, 355, 356, 375, 380–382, 388, 389, 391, 396, 397, 411
Girsanov–Cameron–Martin formula, 362 Gradient Brownian systems, 318, 338 Greens region, 365, 366, 368–371 H Hamilton–Jacobi
equation, 392, 394, 395 function, 392
Harmonic forms, 367–371, 374, 378 Heat kernel, or density, 280, 364, 379–406 Hodge theorem, 367, 371–374, 407 Horizontal lift, 299–305, 310, 311, 319,
326, 335, 362, 383, 386, 404, 411, 412process, 299, 311, 362
Horocycles, 355 H-transform, 351, 366, 370, 377, 378, 391,
395 Hyperbolic space, 312–316, 326, 348, 352,
371, 380, 390, 406 I Injectivity radius, 374 Integration by parts, 338 Invariant measure, 331, 333, 336, 337, 341 Isometry, 313, 316, 342, 350, 353, 354 Ito formulae, 320 J Jacobi field, 327, 328, 330, 358, 381 L Laplace operator
eigenvalue, 364 on forms, 363, 368, 369 spectrum, 363, 364
Left invariant S.D.S., 316–319 vector field, 296, 313, 316, 326
Levi-Civita connection, 306, 307, 314, 315, 317, 320, 321, 324, 326, 327, 399
Lie algebra bracket, 296, 324 group, 296, 316–319, 355, 391
Lifetime, 287 Local
coordinates, 305–307, 383, 399 representatives, 300, 301, 306 trivializations, 297
Lorentz group, 314, 350, 353, 354 Lyapunov
exponents, 340, 348, 357 spectrums, 331–335, 342
M Manifold, 282–284 Maximal solution, 285–287, 292, 294,
308 Mean curvature, 319 Mean exponents, 334–339 Minimal semigroup, 363, 379, 385 Moment exponents, 281, 356–359 N Non-degenerate S.D.S., 304 Normal bundle, 284 Normal coordinates, 301, 303, 319, 350,
395, 396, 399, 401, 404, 406, 409, 412
O One form, 323, 367, 369, 371, 374
heat flow of, 367, 369 Orientable, 312, 342 Orthonormal frame bundle, 308, 312,
342 P Parallel translation, 280, 305, 310, 311,
319, 320, 335, 361 Path lifting property, 347 Pole, 312, 334, 380, 382, 385, 388–390,
395, 397, 403, 405, 409, 412 Principal
bundle, 288, 296–330 curvatures, 319, 321, 341 directions, 319
Projective space, 283, 284
261
R Ricci curvature, 294, 321, 322, 345, 359,
363, 367, 369, 374, 378 Riemannian
connection, 305, 306, 311, 314, 360 cover, 347, 348, 380 distance, 382, 385 metric, 304–308, 314, 316, 320, 321,
332, 334, 341, 347, 349, 374, 380–382, 393
Rolling, 312 Ruse’s invariant, 382, 389 S Second fundamental form, 316–319, 340 Sectional curvature, 322, 374, 380, 390 Semi-martingale, 282, 285, 289, 290, 296,
299, 311 Sphere, 285, 296, 312, 319, 326, 335,
338–340, 356, 380, 381, 390 Stable manifold, 334, 348, 349, 352, 354,
355 Standard Brownian motion with drift,
351 Stochastically complete, 322, 357, 360,
403 Stochastic development, 307–316 Stochastic mechanical flows, 334 Stochastic parallel translation, 311 Strictly conservative, 294 Strongly complete, 294, 295, 327 Structure equations, 329
Submanifold, 281, 283, 284, 286, 287, 304, 316, 317, 321, 324, 343, 352, 380
Supertrace, 407–408, 411 Symmetric space, 315, 335, 355 T Tangent
bundle, 283, 297, 298, 327, 356 space, 283, 284, 296, 297, 304, 309,
312, 319, 324, 341, 349 vector, 283, 304, 325, 353
Tensor bundle, 301, 356, 360, 367, 399 Torsion
form, 323–326 free, 299, 301, 303, 306, 321, 325, 328 tensor, 301, 325
Trivial bundle, 356, 360 Tubular neighbourhood, 284, 286, 290 V Vector field, 287, 288, 296, 300, 302, 303,
305–308, 310, 313, 316–321, 323–329, 334, 336, 362, 367, 381, 397, 400, 401
along a path, 305, 310, 327, 328, 381 Vertical tangent bundle, 297 Volume element, 331, 382 W Weitzenbock formula, 280, 363, 367–369,
398–402 W.K.B. approximation, 394