Space regularity of stochastic heat equationsdriven by irregular Gaussian processes

Oana MocioalcaDepartment of Mathematics

Kent State University, P.O. Box 5190Kent, OH, 44242, USAoana@math.kent.edu

Frederi ViensDepartment of Statistics

Purdue University, 150 N. University St.West Lafayette, IN 47907-2067, USA

viens@purdue.edu

Abstract

We study linear stochastic evolution equations driven by in�nite-dimensional Gaussian processes, some of which are more irregular intime than fractional Brownian motion (fBm) with any Hurst parame-ter H, while others comparable to fBm with H < 1

2 . Sharp necessaryand su¢ cient conditions for the existence and uniqueness of solutionsare presented. Specializing to stochastic heat equations on compactmanifolds, especially on the unit circle, sharp Gaussian regularity re-sults are used to determine su¢ cient conditions for a given �xed func-tion to be an almost-sure modulus of continuity for the solution inspace; these su¢ cient conditions are also proved necessary in highlyirregular cases, and are nearly necessary (logarithmic corrections aregiven) in other cases, including the Hölder scale.

1 Introduction

This article deals with precise existence results for stochastic PDEs drivenby arbitrary Gaussian processes, and specializes to stochastic heat equationsfor sharp spatial regularity results. Since the pioneering work of stochasticanalysts in the 1970�s and 1980�s (see for instance John Walsh�s Saint-Flourlecture notes [24] or DaPrato and Zabczyk�s textbook [6]), probabilists haveinvestigated the question of how to de�ne the weakest conditions su¢ cientto guarantee existence and/or regularity of a stochastic PDE�s solution. Inorder to express results that are as sharp as possible, the choice was made bymany �including our past and present work �to study the simplest possibleproblems with some appeal for applications, hence the use of the stochasticheat equation driven by additive noise. The framework of Ito stochasticcalculus was deemed most appropriate, implying the study of equations ofthe form

u (t; x) = u (0; x) +

Z t

0

�xu (s; x) ds+W (t; x) (1)

for all t � 0, and all x in some (e.g. Euclidean) space E, where W is somerandom �eld on R+ � E. For many years, attention was directed towardsthe case where W is Brownian motion in its parameter t, and it had beenknown since Walsh that W need not be a bona�de function for u to exist,as indeed it may be white noise in space while still allowing a solution in ananalytically weak form (evolution form of DaPrato and Zabczyk), namely forE = R, with p (t; x) = (2�t)�1=2 e�x

2=(2t),

u (t; x) =

ZE

p (t; x� y)u (0; y) dy +ZE

Z t

0

p (t� s; x; y)W (ds; dy) :

In the 1990�s the question of precisely how the spatial regularity of W(or lack thereof) e¤ects the solution�s was posed. For instance, it was foundthat the so-called space-time white noise of the example above yields a so-lution with as much regularity in x as Brownian motion, so that more irreg-ular noises should still imply existence, while qualitatively di¤erent resultsshould be expected in higher space dimensions. Su¢ cient conditions were es-tablished for various additive and non-linear multiplicative stochastic PDEs:[5], [16], [17], [18], [22]. Some of these papers also covered the issue of spa-tial Hölder continuity of the solution, with [22] being the �rst one to supplynecessary and su¢ cient conditions for this property, and its follow-up work

2

[23] providing an indication that extensions to non-Hölder regularity may bepossibleAs the case of Brownian-based noise for stochastic PDEs was now better

understood, the year 2000 saw the emergence of several works focusing on thee¤ect of fractional-Brownian-based models. These are Gaussian noise termsW (t; x) = BH (t; x) whose behavior in time has non-independent incrementsand a scaling property in the power H 2 (0; 1), not simply the Brownian caseH = 1=2; these are simply in�nite-dimensional analogues of scalar fractionalBrownian motion (fBm) introduced e.g. in [13]. The di¢ culty of stochasticintegration with respect to such behavior in time (see [7], [1], [2]), made it sothat not much progress was possible in the case of nonlinear equations: [8],[10], [11], [14]. For linear equations, however, following the impetus in [22]and [23], necessary and su¢ cient conditions for existence were established ina wide abstract setting in [20], and, using the stochastic heat equation onthe circle, for any scale of regularity, Hölder or not, in [21].Still, the issue of changing the time regularity of the driving noise beyond

the fractional Brownian scale has never been addressed. Only recently, in [3]and [4], has the technique for stochastic calculus for highly irregular fBm (lowparameter H) been perfected. Stochastic calculus with respect to arbitrarilyirregular Gaussian processes was performed in [15].Using in�nite dimensional analogues of the processes de�ned in [15], we

follow the framework in the work [20] to �nd necessary and su¢ cient con-ditions for existence of solutions to in�nite-dimensional stochastic evolutionequations driven by these arbitrary Gaussian �elds. Our new calculationtechnique appears to be superior to that employed originally in [20], becausewe establish our existence and uniqueness theorem without needing to as-sume the existence of a spectral gap and �nite-dimensional kernel, whichwas used in [20]. Thus our corresponding work appears as an improvementover, as well as a generalization of, the results in [20].We apply the techniques in [21] for sharp Gaussian regularity in order to

�nd necessary and su¢ cient spatial regularity conditions for the stochasticheat equation on the circle driven by arbitrary Gaussian �elds. These verysharp results are owed to the use of Gaussian �elds with the correspondingregularity theory introduced by Dudley, Fernique, and Talagrand, which werecarefully exploited in our context in [21].Our article is structured as follows. Preliminaries, and the de�nition of

in�nite dimensional Gaussian processes (�elds) with arbitrary regularity isgiven in Section 2. Section 3 gives our existence and uniqueness results.

3

Section 4 gives conditions for speci�c spatial regularity. While the existenceresult of Theorem 4 is stated in an abstract Hilbert-space setting, we summa-rized it here, together with our regularity results, using the stochastic heatequation on the unit circle S1, for illustrative purposes.

Hypothesis Let B (t; x) be a centered Gaussian random �eld de�ned forall (t; x) 2 R+ � S1. Assume B is homogeneous in x, with canoni-cal metric E

�(B (1; x)�B (1; y))2

�= �2 (x� y). Let qn be the nth

Fourier coe¢ cient of �2. Note that if qn is not summable, the abovede�nition is only formally a function, and B must be understood asbeing a Schwartz-distribution-valued Gaussian process. Assume B �sbehavior in time is bounded above as follows:

E�(B (t; 0)�B (s; 0))2

�� 2 (jt� sj)

where 2 is increasing and concave on R+, and di¤erentiable except at0, with (0) = 0. The necessary conditions below are valid speci�callyif B (t; 0) can be written as

R t0

pd 2=dt (t� s) dB (s) where B is a

standard Brownian motion. The su¢ cient conditions do not requirethis form.

Conclusion: existence [Theorem 4]. The stochastic heat equation (1)with W = B has a unique evolution solution in the sense of DaPratoand Zabczyk, in L2 (�R+ � S1), if and only if

1Xn=1

qn 2�n�2�<1:

Conclusion: regularity [Theorem 9]. Let f be an increasing continuousfunction on a neighborhood of 0 with f (0) = 0, di¤erentiable exceptat 0. Let

�f (r) =

Z r

0

f 0 (s) (log (1=s))�1=2 ds

The aforementioned solution admits f , up to a non-random constant,as a uniform modulus of continuity almost surely, if

1Xn=1

qn 2�n�2�h

��2f

�1

n

��<1

4

for all decreasing continuous functions h on a neighborhood of 0 suchthat

R0h (r) dr <1. This is also an �only if�statement when f (r)�

rH for all H > 0. Otherwise, the �only if� part holds with �f (n�1)replaced by �f (n�1) log n.

The su¢ cient conditions for existence and regularity hold for processesde�ned on R+ �R as well, if one simply replaces series by integrals above.Similar su¢ cient condition results also holds in higher dimensions (Rd andother compact or non-compact manifolds). We leave exact statements andproofs of these facts out of this article.

2 Preliminaries

2.1 Irregular Gaussian processes

In the remainder of the article, the symbol � denotes commensurabilitybetween two functions: f and g are commensurable if there exist positiveconstants c; C such that cg (x) � f (x) � Cg (x) for all values of a commonvariable x.A continuous centered Gaussian process X on R+ that starts from the

origin at time 0 has a distribution entirely determined by its increments�variance structure, i.e. the canonical metric

�2 (s; t) = E�(X (t)�X (s))2

�:

The case of Brownian motion X = W is �2 (s; t) = jt� sj, for fBm X =BH we have �2 (s; t) = jt� sj2H . It is well-known that, beyond the scalingproperty of fBm by which BH (ct) = cHt in distribution, fBm admits thefunction f (r) = r log1=2 (1=r) up to a constant as a uniform modulus ofcontinuity almost surely. The so-called Volterra representation of fBm fromstandard Brownian motion has the form

BH (t) =

Z t

0

K (t; s) dW (s) ; (2)

where the kernel K has the property that, for s away from 0, K (t; s) �jt� sjH�1=2.We consider a class of Gaussian processes with arbitrary correlation be-

tween increments, by assuming that a Voterra-type representation holds,

5

where the kernel K can be chosen to be commensurate with some givenfunction that implies a certain type of almost-sure modulus of continuity forBH . To simplify the presentation, and as the only simple means we havefound to ensure that our su¢ cient conditions are also necessary, we assumethat K depends only on the di¤erence t � s. All our su¢ cient conditionsproved below hold for any other K bounded above by a given K (t� s); theproof of this fact is left to the reader. The fact that rH�1=2 =

�d�r2H�=dr�1=2

is a motivation for the de�nition that follows. Also note that any Gaussianprocess starting from 0 that is adapted to a Brownian �ltration must be ofthe Volterra form (2) for an appropriate function K.Let W be a standard Brownian motion on R+ with respect to the prob-

ability space (;F ;P) and the �ltration fFtgt�0. Assume 2 is of class C2everywhere in R+ except at 0 and that d 2=dr is non-increasing. In [15] itis proved that the centered Gaussian process

B (t) :=

Z t

0

" (t� s) dW (s) (3)

where

" (r) :=

�d ( 2)

dr

�1=2: (4)

satis�es the following conditions with respect to fFtgt�0: for any t � 0

(i) � (s; t) � (jt� sj), where � is the canonical metric of B on (R+)2

(ii) B (0) = 0:

(iii) B is adapted to a fFtgt�0.

The choice for B in (3) above is the simplest choice satisfying conditions(i), (ii), and (iii) in terms of applications to stochastic calculus. Again, sinceany process written as

R t0K (t; s) dW (s) withK (t; s) � (jt� sj) will satisfy

the su¢ cient condition statements in our theorems below, our work actuallycovers a very wide class of Gaussian processes, and in particular reaches alongthe entire regularity scale of Gaussian processes.The condition that d 2=dr is non-increasing implies that B is more ir-

regular than standard Brownian motion, such as fBm with H < 1=2, orspeci�cally " (r) = rH�1=2, which yields the so-called Liouville process, whoseregularity and scaling properties are identical to those of fBm, but which has

6

slightly inhomogeneous increments. The case where B is more regular thanBrownian motion, such as fBm with H > 1=2, uses a slightly di¤erent, andconsiderably easier, calculation. We omit this case. In all cases, condition(i) above implies that up to a constant, f (r) = (r) log1=2 (1=r) is a uniformmodulus of continuity for B almost surely, when lim0 f = 0. When thislimit is not 0, one can prove using Gaussian supremum estimation that B

is almost-surely discontinuous everywhere. Nevertheless, all that we claimbelow still holds in this extremely irregular case.

2.2 The Wiener integral with respect to B

Let (B (t))t2[0;T ] be the centered Gaussian process de�ned by its Wienerintegral Volterra-type representation as in (3) and let f be a deterministicmeasurable function on R+:We de�ne the operator K� = K�

acting on f by

K� f (r) :=

�f (r) " (T � r) +

Z T

r

(f (s)� f (r)) "0 (s� r) ds�:

if it exists. If K� f (�) is in L2 ([0; T ]; dr) then we say that f belongs to the

space L2 ([0; T ]), and we denote

kfk2 = K�

f 2L2([0;T ])

=

Z T

0

����f (r) " (T � r) + Z t

r

(f (s)� f (r)) "0 (s� r) ds����2 dr:

This L2 is the so-called canonical Hilbert space of B on [0; T ]. We will also

denote it by H. For any f in H we de�ne the stochastic integral of f withrespect to B on [0; T ] as the Gaussian random variable given byZ T

0

f (t) dB (t) =

Z T

0

K� f (r) dW (r) :

Remark 1. One easily sees that if g is a function in L2 then the functionf : r 7! g (r) 1[0;t] (r) is in L2 and

K�f (r) = 1[0;t] (r)

�" (t� r) g (r) +

Z t

r

[g (s)� g (r)] "0 (s� r) ds�;

i.e. K�f depends on t, not on T .

7

2.3 In�nite dimensional irregular Gaussian processes

We de�ne an in�nite dimensional Gaussian process using the classical ap-proach of [6] for cylindrical Brownian motion (see section 4.3.1 therein). LetU be a real separable Hilbert space. For any �xed complete orthonormalbasis (en)n in U and any �xed sequence of positive numbers (�n)n, even ifP

n�0 �n =1, we de�ne

B (t) =

1Xn=0

p�nenB

n(t); (5)

where B n are an IID sequence of Gaussian processes with the same distrib-ution as the B in the previous section. This slight abuse of notation willnot lead to confusion, since henceforth B denotes an in�nite-dimensionalprocess. Observe that for any �xed t the above series converges in L2(�U)if and only if

Pn�0 �n < 1. In the other case, B (t) will be a well de�ned

Gaussian process with values in a larger Hilbert space U1, where the embed-ding U � U1 is continuous, Hilbert-Schmidt. For instance, if U = L2 ([0; 1]),we can use the space of Schwartz tempered distributions for U1.To de�ne the Wiener integral with respect to the above in�nite dimen-

sional Gaussian process B we consider another real separable Hilbert spaceV and (�s)s2T a deterministic function with values in L2(U ;V ), the space ofHilbert-Schmidt linear operators from U to V . The stochastic integral of �with respect to B is de�ned byZ t

0

�sdB (s) =

1Xn=0

Z t

0

�sendB n(s) =

1Xn=0

Z t

0

(K��en)sdBn(s)

where Bn is the standard Brownian motion used to represent B in theVolterra-type representation (2). The above sum is �nite almost surely, andis indeed a Gaussian random element of V , if and only if it has a �nitevariance in V , i.e. X

n

kK�(�en)k2L2([0;T ];V ) <1:

8

3 Linear stochastic equations. Existence ofsolutions

Consider the setting from the previous section whereB a cylindrical Gaussianprocess (a process de�ned as above for which �n = 1), and � a linear operatorin L(U; V ) that is not necessarily Hilbert-Schmidt.We will study the existence of solutions for the equation

dX(t) = AX(t)dt+ �dB (t); (6)

with boundary condition X(0) = x 2 V , where A : Dom(A) 2 V ! V is thein�nitesimal generator of the strongly continuous semigroup (etA)t2T , a selfadjoint operator on V .

Remark 2. This equation is de�ned if the integralR t0�dB is de�ned as a

V -valued random variable. Since

E

����Z t

0

�dB (s)

����2V

=Xn

E

����Z t

0

�endB n(s)

����2V

=Xn

E

����Z t

0

dB n(s)

����2 j�enj2V = 2(t)k�k2HS;this occurs if and only if � 2 L2(U; V ).Nevertheless, it is standard practice in stochastic analysis, useful in ap-

plications, to consider a weak form of the above di¤erential equation, well-de�ned even for many non-Hilbert-Schmidt integrands �, depending on theregularity of the operator A.

De�nition 3. The mild, or evolution, form of the stochastic di¤erentialequation (6), with starting point X (0) = x 2 V , is given as follows: for allt 2 [0; T ], we have the equality in V almost surely:

X(t) = etAx+

Z t

0

e(t�s)A�dB (s):

Since this is an explicit formula for X, we call it the mild, or evolution,solution to (6).

9

Reed and Simon in [19] presented the existence of an uniquely de�nedprojection measure dP� on the real line, such that for every � 2 V , dh�; P��iis a Borel measure on R, and for every � 2 Dom(A)

h�;A�i =ZR�dh�; P��i:

The next theorem is a generalization of Theorem 40 in [15].

Theorem 4. Let " and be de�ned as in Section 2 (see de�nitions (3) and(4)). Assume

j"0(x)j"(x)

� 1

2x(7)

andj"0(x)j � x� 3

2f(x) (8)

hold, where f is an increasing di¤erentiable function, and let B be a cylindri-cal Gaussian process on the Hilbert space U . Assume A : Dom(A) 2 V ! Vis a negative, self-adjoint operator on the Hilbert space V . Note that we donot need to assume A is negative-de�nite.Then for any �xed � 2 L(U; V ) there is a unique mild solution X(t) 2

L2(; V ) for the equation

dX(t) = AX(t)dt+ �dB (t); t 2 [0; T ]; X(0) = x 2 V

if and only if ��GH(�A)� is a trace-class operator, where

GH(�) = 2

�1

max(�; 1)

�:

Here for any integrable function F onR+, F (�A) is de�ned by F (�A)x =R10F (�)P��xd� for all x 2 V .

Remark 5. Our conditions on " are not restrictive. Since "2 is integrable near0 due to de�nition (3), we can assume without loss of generality that "2 (x)xis increasing near 0. This means 2" (x) "0 (x)x+ "2 (x) � 0, which implies (7)for all x near 0.

Remark 6. Condition (8) signi�es how to compare the Volterra kernel withthose standard power scale ones, with (x) = xH , so that j"0 (x)j = cHx�3=2xH ,or logarithmic scale ones, with (x) = log�� (1=x), so that indeed j"0 (x)j �x�3=2 log�� (1=x), from which we see that requiring f increasing is not arestriction in any scale.

10

Proof of the theorem.Step 1: setup and exact calculations.Consider the scalar measure �n de�ned as

d�n(�) = dh�en; P��eniV :

DenotingIt = EjX(t)� etAxj2V ;

it is su¢ cient to estimate It optimally from above and below. By indepen-dence of the components in the de�nition (5) of the in�nite-dimensional B ,we have

It = EjX(t)� etAxj2V = E����Z t

0

e(t�s)A�dB (s)

����2V

= E

�����Xn

Z t

0

e(t�s)A�endB n(s)

�����2

V

Then using the de�nition of Wiener integration in Section 2.3,

It = E

�����Xn

Z T

0

K� �e(t�s)A�en1[0;t](s)� dW (s)�����2

V

=Xn

Z T

0

��K�

�1[0;t]e

(t�s)A�en���2Vds

=Xn

Z t

0

����e(t�s)A�en"(t� s) + Z t

s

(e(t�r)A�en � e(t�s)A�en)"0(r � s)dr����2V

ds

For ease of computations we can take t = 1. Then, by de�nition of esA,

I1 =Xn

Z 1

0

����e(1�s)A�en"(1� s) + Z 1

s

(e(1�r)A�en � e(1�s)A�en)"0(r � s)dr����2V

ds

=Xn

Z 1

0

Z 1

0

�e�(1�s)�"(1� s) +

Z 1

s

(e�(1�r)� � e�(1�s)�)"0(r � s)dr�dsd�n(�)

=Xn

Z 1

0

(Z 1

0

"e�2�(1�s)"2(1� s) +

�Z 1

s

(e��(1�r) � e��(1�s))"0(r � s)dr�2

11

+2e��(1�s)"(1� s)Z 1

s

(e��(1�r) � e��(1�s))"0(r � s)dr�ds

�d�n(�)

=Xn

Z 1

0

I(�)d�n(�) (9)

where

I(�) =

Z 1

0

"e�2�(1�s)"2(1� s) +

�Z 1

s

(e��(1�r) � e��(1�s))"0(r � s)dr�2

+2e��(1�s)"(1� s)Z 1

s

(e��(1�r) � e��(1�s))"0(r � s)dr�ds

=

Z 1

0

e�2�(1�s)

("2(1� s) +

�Z 1

s

(e�(r�s) � 1)"0(r � s)dr�2

+2"(1� s)Z 1

s

(e��(r�s) � 1)"0(r � s)dr�ds

=

Z 1

0

e�2�(1�s)�"(1� s) +

Z 1

s

(e�(r�s) � 1)"0(r � s)dr�2ds

=

Z 1

0

e�2�(1�s)�"(1� s) +

Z 1�s

0

(eu� � 1)"0(u)du�2ds

=

Z 1

0

e�2�s�"(s) +

Z s

0

(eu� � 1)"0(u)du�2ds: (10)

The equality before the last one is obtain by the change of variable u = r� sand the last one with the change of variable s = 1 � s. Estimation of theformula in (10) is non-trivial: indeed the two terms in the square inside I (�)are of opposite signs, since " (r) =

pd 2=dr is assumed to be decreasing. In

steps 2 and 3 below, we assume � � 1.Step 2: upper bound when � � 1.We �rst establish an upper bound on I (�). We observe that

I(�) � 2"Z 1

0

e�2�s"2(s)ds+

Z 1

0

e�2�s�Z s

0

(eu� � 1)"0(u)du�2ds

#= I1(�; 1) + I2(�; 1)

This is not a sharp inequality a priori, since it kills the negativity of "0.Nevertheless, our lower bound below shows that it is actually sharp.

12

Step 2a: upper bound, 2nd term. We start by analyzing the second termof the inequality

I2(�; 1) =

Z 1

0

e�2�s�Z s

0

(eu� � 1)"0(u)du�2ds

�Z 1

�

0

e�2�s�Z s

0

(eu� � 1)"0(u)du�2ds

+

Z 1

1�

e�2�s2

Z 1�

0

(eu� � 1)"0(u)du!2ds

+

Z 1

1�

e�2�s2

Z s

1�

(eu� � 1)"0(u)du!2ds

:=I2;0(�) + I2;1(�) + I2;2(�);

and now bounding each of these three terms. To control the �rst term webound e�r�1 above by C�r and 2�2�s by 1. The actual value of the constantC below may change from line to line, but never depends on �.

I2;0 (�) � CZ 1=�

0

�2�Z s

0

r j"0 (r)j dr�2ds:

Keeping in mind the special condition (7) we obtain

I2;0(�) �C�2Z 1

�

0

�Z s

0

"(r)dr

�2ds

�C�2Z 1

�

0

Z 1�

0

"(r)dr

!2ds

=C�

Z 1�

0

"(r)dr

!2For the second term, using the same approximations and inequality as

above, we obtain

I2;1 (�) � CZ 1

1=�

e�2s�

Z 1=�

0

�r j"0 (r)j dr!2ds

13

� CZ 1

1=�

e�2s�

�

Z 1=�

0

" (r) dr

!2ds

= C1

2�

�e�2 � e�2�

� �

Z 1=�

0

" (r) dr

!2

� Ce�2

2�

Z 1�

0

"(r)dr

!2

= C�

Z 1�

0

"(r)dr

!2The last term can be evaluated as in [15]. It was shown that if "0 has the

representation j"0 (r)j � r�3=2f (r) with f di¤erentiable and increasing, andj"0j decreasing, then

I2;2 � Cf 2(1

�):

We can rewrite this as

I2;2 � Cf 2(1

�) = C�3��3f 2(

1

�)

= C1

�3

�1

�

�� 32

f(1

�)

!2� C 1

�3("0)2(

1

�)

= C1

�

1

�2("0)2(

1

�) � C 1

�"2(1

�)

= C�

�1

�"(1

�)

�2� C�

Z 1�

0

"(r)dr

!2where the last inequality was obtain by the monotonicity of ".Putting the bounds of the three terms together we obtain

I2(�; 1) � I2;0(�) + I2;1(�) + I2;2(�)

� C� Z 1

�

0

"(r)dr

!2

14

Step 2b: upper bound, �rst term. For the upper bound of the �rst termin the evaluation of I(�), I1(�; 1), we use the fact that for s � 1

�we have

e2�s � 1(�s)2

, we use a scalar change of variables and the monotonicity of " inorder to obtain

I1(�; 1) =

Z 1

0

e�2�s"2(s)ds �Z 1

�

0

"2(s)ds+

Z 1

1�

1

(�s)2"2(s)ds

=

Z 1�

0

"2(s)ds+1

�

Z �

1

1

u2"2(u

�)du

�Z 1

�

0

"2(s)ds+1

�"2(1

�)

Z �

1

1

u2du

�Z 1

�

0

"2(s)ds+1

�"2(1

�)

�Z 1

�

0

"2(s)ds+ �

Z 1�

0

"(s)ds

!2:

Again using the assumption

j"0 (x)j" (x)

� 1

2x:

we have Z x

0

" (r) dr � 2Z x

0

j"0 (r)j rdr = �2" (x)x+ 2Z x

0

" (r) dr

which impliesR x0" (r) dr � 2" (x)x and in particular

�

Z 1�

0

" (r) dr � 2"�1

�

�:

Then using the monotonicity of "

2(1

�) =

Z 1�

0

"2(r)dr � "( 1�)

Z 1�

0

"(r)dr

� C� Z 1

�

0

"(r)dr

!2

15

This completes the upper bound proof since, now

I(�) � 2( 1�) + C�

Z 1�

0

"(r)dr

!2� 2( 1

�) + C 2(

1

�) = C 2(

1

�)

Step 3. Lower bound when � � 1.For the lower bound there is a simple strategy. It is certainly true that

1= (2�) � 1, and from (10) we obtain the trivial lower bound

I(�) �Z 1

2�

0

e�2�s�"(s) +

Z s

0

(eu� � 1)"0(u)dr�2ds:

Since the two terms inside the square are of opposite sign, our strategy isto show that the second term (in absolute value) is less than a constant Ktimes the �rst term, with K < 1: if we haveZ s

0

(eu� � 1)j"0(u)jdr � K"(s) (11)

then

"(s)�Z s

0

(eu� � 1)j"0(u)jdr � (1�K)"(s)

which implies, using the fact that " decreases,

I(�) �Z 1

2�

0

e�2�s ((1�K)"(s))2 ds

= (1�K)Z 1

2�

0

e�2�s"2(s)ds

� (1�K) e�1Z 1

2�

0

"2(s)ds � (1�K) e�1Z 1

2�

0

"2(2s)ds � (1�K) e�1Z 1

�

0

"2(s)ds

= (1�K) e�1 2( 1�);

which is all that is needed for the proof of the lower bound.To establish this we use again the special condition (7). Since "2 is in-

tegrable at the origin, it holds that "2 (r) = o (r�1) and thus we can also

16

assume without loss of generality, similarly to condition (8) (see Remark 6),that there exists an increasing function g with with "(r) = r�

12 g(r). Thus

we get Z s

0

(eu� � 1)j"0(u)jdr � �

2

Z s

0

eu� � 1u�

"(u)du

Since the function ex�1xis bounded for x 2 [0; 1

2], by 1 below and

pe�12above,

we getZ s

0

(eu� � 1)j"0(u)jdr � �pe� 12p2

Z s

0

"(u)du

= �

pe� 12p2

Z s

0

u�12 g(u)du � �

pe� 12p2g(s)2

ps

=

pe� 1p2�s"(s)

�pe� 12p2"(s):

Sincepe�12p2< 1 this completes the proof for the lower bound when � � 1.

Step 4: conclusion when � � 1.From the results of Steps 2 and 3, we have proved that for any � � 1,

I (�) � 2( 1�) = 2

�1

max (�; 1)

�:

Step 5: case � 2 [0; 1].A precise estimate of I (�) is more di¢ cult in this case, but we do not

need to have a precise result. Indeed, we only need to show that for all� 2 [0; 1],

I (�) � 2�

1

max (�; 1)

�= 2 (1) :

In other words, we only need to show that I (�) is bounded above and belowby positive constants, uniformly in � 2 [0; 1].Using ex � 1 � ex for x 2 [0; 1], using "0 < 0, and integrating by parts

(using the fact that " (s) = o�s�1=2

�), the negative term (with the "0) in

formula (10) is bounded above as����Z s

0

�eu� � 1

�"0 (u) du

����2 � e2 ����Z s

0

u�"0 (u) du

����217

= e2�2����Z s

0

(" (u)� " (s)) du����2

� e2s�2Z s

0

"2 (u) du = e2s�2 2 (s) :

where we used Jensen�s inequality in the last step. Therefore we immediatelyhave the upper bound

I (�) �Z 1

0

2"2 (s) ds+

Z 1

0

2e2s�2 2 (s) ds

��2 + 2e2

� 2 (1) :

For a lower bound, de�ne

f (�; s) = "(s) +

Z s

0

(eu� � 1)"0(u)du: (12)

From (10), we see that I (�) � e�2R 10f 2 (�; s) ds. A positive lower on I (�)

uniform for all � 2 [0; 1] now follows from the lemma below.Step 6: �nal lemma, and conclusion.Since the results above, including the next lemma, establish that I (�) �

2 (1=max (�; 1)) for all � 2 R+, the theorem follows; indeed we can assert

I1 �Xn

Z 1

0

2 (1=max (�; 1)) d�n(�) = tr (��GH(�A)�) :

�

Lemma 7. With f (�; s) as in (12), minnR 1

0f 2 (�; s) ds : � 2 [0; 1]

o> 0:

Proof. First note that f is di¤erentiable with respect to s everywhereexcept at 0, and that we have

@f

@s(�; s) = "0 (s) es� < 0

so that f (�; �) is decreasing on (0; 1]. We have lims!0 " (s) = +1, and weproved in Step 5 that

lims!0

����Z s

0

�eu� � 1

�"0 (u) du

���� � lims!0 e2s 2 (s) = 0;18

Therefore lims!0 f (�; s) = +1. Hence for each � 2 [0; 1], there exists avalue s� (�) 2 (0; 1] such that f (�; s) � 1 for all s � s� (�), and de�ne s� (�)to be maximal such. Note that for those values of � for which f (�; s) exceeds1 for all s 2 [0; 1], this simply means that the corresponding s� (�)�s are allequal to 1. Moreover, we calculate

@f

@�(�; s) =

Z s

0

ueu�"0 (u) du < 0;

so that s� is non-increasing. This means that, de�ning s�� = s� (1) which isstrictly positive as noted above, we have for all s � s��, for all � 2 [0; 1],f (�; s) � 1, and we �nally obtainZ 1

0

f 2 (�; s) ds �Z s��

0

f 2 (�; s) ds

� s�� > 0;

�nishing the proof of the lemma. �

4 Functional equations and space regularity

4.1 Evolution equations on manifolds

The abstract framework of Theorem 4 is useful in a number of more concretesituations. We will illustrate this point by investigating the so-called additivestochastic heat equation, namely a parabolic stochastic PDE of the form

u (t; x) = u0 (x) +

Z t

0

�xu (s; x) ds+W (t; x) (13)

for some Gaussian random �eldW on the cartesian product ofR+�M whereM is a �nite-dimensional space where �x can be de�ned;W could thus rangeover a wide array of in�nite-dimensional versions of our B (t) de�ned by (3).Using again the evolution interpretation in the manner of Da Prato and

Zabczyk [6], we replace (13) by

u (t; x) = Ptu0 (x) +

Z t

0

Pt�sW (ds; �) (x) : (14)

19

where (Pt)t�0 is the semigroup of operators generated by �x. Indeed, asolution to (13) also solves (14), but the latter is considerably weaker, sincethe Wiener integral above may exist even if W is not a bona�de function inx.More speci�cally, for an arbitrary smooth compact Riemannian manifold

M and its Laplace-Beltrami operator �x, let (�n; en)n2N be the eigenval-ues and eigenfunctions of �x which we can arrange in increasing order with�0 = 0 and �n > 0 for all n > 0; then under the Riemannian inner product,fengn2N can be chosen as an orthonormal basis for a Hilbert space of func-tions V on M , the space of square-integrable functions with respect to theRiemannian volume element.We use for W the random �eld B de�ned formally by

B (t; x) =Xn2N

pqnen (x)B

(t)

where (B n)n2N is a family of independent copies of our B in (3). If

Pn qn is

�nite, this B (t; �) is a V -valued Gaussian element, but ifP

n qn is in�nite,this de�nition is only formal, and in reality B (t; �) is typically distribution-valued. Theorem 4 is easier to express in this framework because �x and thespatial covariance of B are both diagonalizable in the basis of V , and also�x has a spectral gap. We have that

u (t; x) = Ptu0 (x) +

Z t

0

Pt�sB (ds; �) (x) (15)

has a solution in u (t; �) in V if and only ifXn2N

qn 2

�1

�n

�<1; (16)

and in this case the solution is a Gaussian random �eld on R+ �M , and isalso a V -valued Gaussian stochastic process. Since limr!0 (r) = 0, thereare obviously solutions of (15) corresponding to non-summable sequences qn,i.e. to �elds B which are not bona�de L2 functions in x. This is the usualobservation for additive stochastic PDEs, generalized here to all scales ofpotential irregularity in time, and there seems to be little to gain by singlingout the case of summable qn.The above development works also for non-compact manifolds, such as

Rd, where all above statements can be written using integrals with respect

20

to n2 = � 2 R+ instead of series over n 2 N, but the absence of aspectral gap for the Laplacian forces one to revert to expressions involving 2 (1=max(�; 1)) instead of simply 2 (1=�). We leave these details out.Returning to the compact case, if condition (16) is satis�ed more than

just barely, one should expect some regularity for the solution in (15). Wenow illustrate this phenomenon in the speci�c example of the circle.

4.2 Solution regularity on the unit circle

Assume now thatM = S1, the circle parametrized by [0; 2�). Therefore, it ismost convenient to represent the eigenstructure of the Laplace-Beltrami �x

by saying that for each n 2 N the eigenvalue �n = �n2 has two eigenfunctionsen (x) = cos (nx) and fn (x) = sin (nx). Let now B be as above, andassume in addition that it is homogeneous in the parameter x, a with a givencovariance structure Q in space, which means that it can be represented as

B (t; x) =

Z t

0

" (t� s)W (ds; x)

where the Gaussian �eldW onR+�S1 has covariance E [W (t; x)W (s; y)] =Q (x� y)min (s; t). It is then possible to express the decomposition of B inthe trigonometric basis of V = L2 (S1), i.e. as a Gaussian Fourier series:

B (t; x) =pq0B

0 (t)+

1Xn=1

pqn �B

n (t) sin (nx)+

1Xn=1

pqnB

n (t) cos (nx) (17)

where (B n)n and��B n�nare independent families of independent copies of

the B in (3), and (qn)n is a sequence of non-negative numbers. In fact,since Q is a positive de�nite function on S1, and thus a member of L1 (S1),the values qn are easily seen to be its Fourier coe¢ cients. Since sin (nx) andcos (nx) share the eigenvalue exp (�n2t) for Pt, we can immediately rewrite(15), assuming without loss of generality that u0 � 0, that

u (t; x) =pq0

Z t

0

e�(t�s)n2

dB 0 (s) +1Xn=1

pqn cos (nx)

Z t

0

dB n (s) e�(t�s)n2

(18)

+1Xn=1

pqn sin (nx)

Z t

0

d �B n (s) e�(t�s)n2

21

with, by Theorem 4, existence and uniqueness holding if and only ifXn2N

qn 2�n�2�<1: (19)

More precisely, since the proof of Theorem 4 translates here as nothingmore than an estimation of the variances of the Gaussian random variablesin (18) from above and below, and (18) clearly shows that u (t; �) is spatiallyhomogeneous, we have actually proved the following.

Corollary 8. When V = L2 (S1), with the V -valued Gaussian process B

de�ned in (17), and u the solution (15) of the stochastic heat equation drivenby B , expressed for example in (18), for every �xed t 2 [0; 1], u (t; �) is ahomogeneous Gaussian �eld on S1 satisfying

u(t; �) =Xn2Z

psn(t)

�cos (n; �)Gn + sin (n; �) �Gn

�where G and �G are independent sequences of IID standard normal randomvariables, and

sn(t) � qn 2(n�2):The commensurability constants depend on t, and the sequence fqngn2N,but not on n.

This corollary is all that is needed to apply the regularity results in Section3.3 of [21]. We leave the proof of the theorem below, which is no morethan bookkeeping, to the reader. It gives su¢ cient, and largely necessary,conditions for the solution u to have a prescribed almost-sure modulus ofcontinuity in space. Let Y be the Gaussian random �eld de�ned on S1 by

Y (x) =pq0Z0 +

1Xn=1

pqn

�n�2� �Zn sin (nx) + �Zn cos (nx)

�(20)

where Z and �Z are independent sequences of IID standard normal r.v.�s.Since Y is clearly a homogeneous Gaussian �eld on S1, we can calculate itshomogeneous canonical metric function �Y :

�2Y (x� y) = E�(Y (x)� Y (y))2

�=

1Xn=1

pqn

�n�2�(1� cos (n (x� y))) :

(21)

22

Consider then the function f = f�Y de�ned on a neighborhood of 0 via therule

f (�) = f� (�) :=

Z 1

0

��min

�e�x

2

; ���dx =

Z �(�)

0

qlog 1=�� (")d"

where �� is the inverse function of �. From the work of Fernique [9], whichinterprets the so-called Entropy upper bound of Dudley (see [12]), we knowthat f is an almost-sure uniform modulus of continuity for Y , i.e. that

sup

�jY (x)� Y (y)jf (jx� yj) : x; y 2 S1

�(22)

is almost-surely �nite. The following theorem, established exactly like The-orem 4 in [21],.shows that f is also an almost-sure uniform modulus of con-tinuity for u (t; �), gives a way to construct a Y and a u (t; �) which share agiven function f as an almost-sure uniform modulus of continuity, by ensur-ing a convergence condition on the coe¢ cients qn, and shows a converse istrue in the sense that if u (t; �) has f as an almost-sure uniform modulus ofcontinuity, then the convergence condition should hold.

Theorem 9. Let f be an increasing continuous function on a neighborhood of0 in R+, continuously di¤erentiable everywhere except at 0, with lim0+ f = 0.Let Y and u be given by (20) and (18). Let �f be given by

�f (�) = f (�) (log (1=�))�1=2 �

Z �

0

f (r) (log (1=r))�3=2 (2r)�1 dr:

Note that no claim regarding whether this function �f has the form in (21)is necessary here. It is, however, positive and increasing.

Su¢ cient Condition Assume that for any continuous, decreasing, di¤er-entiable function h on [0; 1] with

R 10h (x) dx <1;

Xn

qn 2�n�2�h

�f

�1

n

�2!<1: (23)

Then f is an almost-sure uniform modulus of continuity for both Y andu (t; �), in the sense of (22).

23

Necessary condition: sharp case When f (r) � rH for any H > 0, theconverse is true. Namely, assume f is an almost-sure uniform modulusof continuity for Y or for u (t; �); then (23) holds.

Necessary condition: Holder case When it is not true that f (r) � rH

holds for all H > 0, the converse is nearly true, up to a logarithmiccorrection. Namely, assume f is an almost-sure uniform modulus ofcontinuity for Y or for u (t; �); then (23) holds with �f (1=n) replacedby �f (1=n) log (n).

It should be noted that the canonical metrics of Y and u (t; �) are, up toconstants, bounded above by �f .

A similar theorem which, instead of Condition (23), uses the conditionthat Y admits f as an almost-sure uniform modulus of continuity, also holds.See Theorem 3 in [21]. We �nish this article with some examples of theprecision allowed by the above theorem.

4.3 Examples on the unit circle

4.3.1 Fractional Brownian scale

In the case of the fractional Brownian scale, where (r) � rH0, becausethe logarithmic correction is not visible in the Hölder scale, because thecorrection needed to make the function h (r) = 1=r integrable at the originis also logarithmic, and lastly because the ratio of f (r) = rH

0over the

corresponding �f is again in the logarithmic scale, we can state the followingnecessary and su¢ cient condition. The solution u to (19) is almost surelyH 0-Hölder-continuous in x for all H 0 < H1 if and only if, for all H 0 < H1,X

n

qnn�4H0+2H0

<1:

For instance, ifpqn � n�1=2�H

00, we get

H 00 > H1 � 2H0: (24)

To be more precise, including the logarithmic terms, and using a general , to get that u is precisely H1-Hölder continuous, i.e. to get f (r) = rH1, we

24

see that �f (r) � rH1 log�1=2 (1=r), so it is su¢ cient to choose qn such that(using h (r) = r�1 log�1 (1=r) (log log)�1 (1=r)),

1 >Xn

qn 2�n�2�n2H1 � log n � log�1

�n2H1 log n

�(log log)�1

�n2H1 log n

��Xn

qn 2�n�2�n2H1 (log log)�1 (n) :

Similarly, to obtain a u which has exactly the same regularity in space asthe fBm with parameter H1, we need f (r) = rH1 log

1=2 (1=r). Thus we get�f (r) � rH1, and we only need to require that,

1 >Xn

qn 2�n�2�n2H1 log�1 (n) (log log)�1 (n) :

Because of the logarithmic correction needed to make the converse work,we can only state, for instance, that if f (r) = rH1 is an almost-sure modulusof continuity for u in space, then for any � > 1,X

n

qn 2�n�2�n2H1 log�2 (n) (log log)�� (n) <1;

and the log�2 n above should be replaced by log�1 n if we only know that uhas the same regularity in space as fBm with parameter H1.

4.3.2 Logarithmic regularity scale

The case of Gaussian �elds whose almost-sure modulus of continuity is com-mensurate with f (r) = log�� (1=r) for � > 0, which we like to call thelogarithmic Brownian scale, coincides, according to our statements regardingY in the above theorem, with �f (r) � log���1=2 (1=r), and coe¢ cients qnsatisfying, up to a triply iterated logarithmic term,X

n

qn 2�n�2�log2� (n) (log log)�1 (n) <1

for some � > 1. More precisely, the above condition is su¢ cient for u to havef (r) = log�� (1=r) as a uniform modulus of continuity in x, but if the latterholds, then the above series converges if one adds a factor (log log log)�� (n)for a > 1.

25

When B itself is in the logarithmic Brownian scale in time, meaning (r) � log��0�1=2 (1=r) for some �0 > 0, we �nd that log��1 (1=r) is a uniformmodulus of continuity for u in x as soon asX

n

qn log2�1�2�0�1 (n) (log log)�1 (n) <1

with the condition being necessary if a factor (log log log)�� (n) is added, andindeed the necessary and su¢ cient condition is simply thatX

n

qnan log2�1�2�0 (n) <1

for all positive sequences fangn2N such that n�1an is summable. For instance,if we assume that

pqn � n�1=2 log�2�

00(n), we see that we only need to take

�00 � �1 � �0: (25)

4.3.3 Conclusion

While the last result may seem esoteric, it actually has an important in-terpretation, when compared to (24). First we have the fact that (25) ismore precise than (24) �we have an exact upper bound on �00; not a gap asrequired in (24). But more importantly condition (24) indicates that to ob-tain a H1-Hölder-continuous solution u, the Hölder-continuity of B in space(measured by H 00 up to an additive constant) has to be combined with B �sHölder-continuity in time (measure by H0), but that the latter is twice asstrong as the former; this is a phenomenon familiar to those who know thatfor the standard stochastic heat equation with in�nite-dimensional Brown-ian potential (here (r) = r�1=2), when the spatial regularity of B is suchthat solution is H-Hölder-continuous in space, then it is only H=2-Hölder-continuous in time. The situation in the logarithmic scale is not the same.Condition (25) shows that the combined logarithmic continuity of B in spaceand time are to be compared with equal weights (�0 + �00), i.e. without thefactor 2 in time, with the solution�s logarithmic continuity. In conclusion,the common intuition saying that the stochastic heat equation�s regularity istwice as strong in space as it is in time, the factor 2 being due to the quadraticvariation of Brownian motion, is misleading. We see here that, in the Hölderscale, the e¤ect of the potential B �s time regularity is always twice as heavy

26

as its space regularity, that this appear to be a general property of the heatequation since it has nothing do to with the presence of white-noise in time,as it holds for all (r) � rH0, not just H0 = 1=2. But on the other hand,the relative strengths of the potential�s time regularity is equal, not double,its time regularity, in the logarithmic regularity scale, which means that thetype of noise can make a di¤erence in the potential�s regularity e¤ect for theheat equation, even though one has to reach to logarithmic regularity to getthis di¤erent e¤ect.

References

[1] Alòs, E.; Mazet, O.; Nualart, D. Stochatic calculus with respect toGaussian processes. Annals of Probability 29 (2001) 766-801.

[2] Alòs, E.; Nualart, D. Stochastic integration with respect to the fractionalBrownian motion. Stoch. Stoch. Rep. 75 (2003), no. 3, 129�152.

[3] Bender, Ch. An Itô formula for generalized functionals of a fractionalBrownian motion with arbitrary Hurst parameter. Stochastic Process.Appl. 104 (2003), no. 1, 81�106.

[4] Cheridito, P. Nualart, D. Stochastic integral of divergence type with re-spect to fractional Brownian motion with Hurst parameter H 2 (0; 1=2).Preprint, 2003. URL: http://orfeu.mat.ub.es/~nualart/cn03c.pdf.

[5] Dalang, R., Frangos, N. The stochastic wave equation in two spatialdimensions. Ann. Proba. 26 (1998) 187-212.

[6] Da Prato, G.; Zabczyk, J. Stochastic equations in in�nite dimensions.Encyclopedia of Mathematics and its Applications, 44. Cambridge U.P., 1992.

[7] Decreusefond, L.; Ustunel, A.-S. Stochastic analysis of the fractionalBrownian motion. Potential Analysis, 10 (1997), 177-214.

[8] Duncan, T.E.; Maslowski, B.; Pasik-Duncan, B. Fractional Brownianmotion and stochastic equations in Hilbert spaces. Stoch. Dyn., 2 (2002),225-250.

27

[9] Fernique, X. Fonctions Aléatoires Gaussiennes, Vecteurs AléatoiresGaussiens. Les publications CRM, Montréal, 1997.

[10] Grecksch , W. ;Ahn, V.V. A parabolic stochastic di¤erential equationwith fractional Brownian motion input. Statistics and Probability Let-ters, 41 (1999), 337-346.

[11] Kleptsyna, M.L.; Kloeden, P.E.; Ahn, V.V. Existence and uniquenesstheorems for stochastic di¤erential equations with fractal Brownian mo-tion. (Russian) Problemy Peredachi Informatsii 34 (1998), no. 4, 51�61;translation in Problems Inform. Transmission 34 (1998), no. 4, 332�341(1999).

[12] Ledoux, M.; Talagrand, M. Probability on Banach Spaces. Birkhauser,1990.

[13] Mandelbrot, B.B.; Van Ness, J.W.; Fractional Brownian motion, frac-tional noises and application. SIAM Review 10 (1968), no. 4, 422-437.

[14] Maslowski, B.; Nualart, D. Evolution equations driven by a fractionalBrownian motion. Preprint, 2002. To appear in J. Functional Analysis.

[15] Mocioalca, O.; Viens, F Skorohod integration and stochastic calculusbeyon the fractional Brownian scale, J. Functional Analysis, vol 222,Issue 2, (2005), 385-434.

[16] Millet, A., Sanz-Solé, M. A stochastic wave equation in two spatial di-mensions: smoothness of the law. Ann. Probab. 27 (1999), no. 2, 803�844.

[17] Peszat, S., Zabczyk, J. Stochastic evolution equations with a spatiallyhomogeneous Wiener process. Stochastic Process. Appl. 72 (1997), no.2, 187�204.

[18] Peszat, S.; Zabczyk, J. Nonlinear stochastic wave and heat equations.Probab. Theory Related Fields 116 (2000), no. 3, 421�443.

[19] Reed, M; Simon, B Functional Analysis I 2nd ed. Academic press, Inc.,1980

28

[20] Tindel, S.; Tudor, C.A.; Viens, F. Stochastic evolution equations withfractional Brownian motion. Probab. Theory Related Fields 127 (2003),no. 2, 186�204.

[21] Tindel, S.; Tudor, C.A.; Viens, F.G.. Sharp Gaussian regularity on thecircle, and applications to the fractional stochastic heat equation. Toappear in Journal of Functional Analysis, 2004.

[22] Tindel, S.; Viens, F. On space-time regularity for the stochastic heatequation on Lie groups. J. Funct. Anal. 169 (1999), no. 2, 559�603.

[23] Tindel, S.; Viens, F. Regularity conditions for parabolic SPDEs on Liegroups. Seminar on Stochastic Analysis, Random Fields and Applica-tions, III (Ascona, 1999), 269�291, Progr. Probab., 52, Birkhäuser,Basel, 2002.

[24] Walsh, J. B. An introduction to stochastic partial di¤erential equa-tions.In: Ecole d�Eté de Probabilités de Saint Flour XIV, Lecture Notesin Math. 1180 (1986), 265-438.

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