SPDEs and regularity structures

SPDEs and regularity structures

Lorenzo Zambotti (LPMA)

Haifa, December 2017

Lorenzo Zambotti Technion, December 2017

Four papers

I Martin Hairer (2014),A theory of regularity structures, Inventiones.

I Yvain Bruned, M.H., L.Z. (2016),Algebraic renormalisation of regularity structures, arXiv.

I Ajay Chandra, M.H. (2016),An analytic BPHZ theorem for regulariy structures, arXiv.

I Y.B., A.C., Ilya Chevyrev, M.H. (2017),Renormalising SPDEs in regularity structures, arXiv.

This quartet of papers "gives a completely automatic black box forlocal existence and uniqueness theorems for a wide class of SPDEs".


Singular stochastic PDEs

Let ⇠ be a space time white noise (or more generally a random elementof D0(Rd), space of generalised functions)

(KPZ) @tu = �u + (@xu)2 + ⇠, x 2 R,

(PAM) @tu = �u + u ⇠, x 2 R2,

(�43) @tu = �u � u3 + ⇠, x 2 R3.

Even for polynomial non-linearities, we do not know how to properlydefine products of (random) distributions.

Note that if T 2 D0(Rd) and 2 D(Rd), then we can definecanonically the product T = T 2 D0(Rd) by

T(') = T (') := T( '), ' 2 D(Rd).


Analogy with stochastic calculus

Similar problem with stochastic integrals.

(Bt)t�0 a Brownian motion, (At)t�0 a smooth process,Z t

0As dBs := AtBt �

Z t

0Bs As ds, t � 0.

Several possible extensions of this definition to a larger class ofnon-smooth A (Itô, Stratonovich...).

For A such that the Itô integral is well defined, B ! R •0 As dBs is

measurable but not continuous.

Starting point of the Rough Paths theory (Terry Lyons, MassimilianoGubinelli).


Wong-Zakai

Let us consider the ODE in Rd

x" = b(x") + f (x") B" (1)

where B" is a smooth approximation of a BM B. Then it is well knownthat x" ! x solution to the Stratonovich SDE

dx = b(x) dt + f (x) � dB.

In order to obtain the Itô SDE in the limit, one has to define rather

ddt

x" = b(x") � 12

Df (x") f (x") + f (x") B" (2)

and in this case x" ! x solution to

dx = b(x) dt + f (x) dB.

Now, (2) is a renormalisation of (1).Lorenzo Zambotti Technion, December 2017

Regularisation

Let ⇠" = %" ⇤ ⇠ a regularisation of ⇠ and let u" solve

@tu" = �u" + F(u", ru", ⇠").

What happens as " ! 0 ?

We need a topology such thatI the map ⇠" 7! u" is continuousI ⇠" ! ⇠ as " ! 0.

For classical negative Sobolev spaces the first point fails.

For classical positive Sobolev spaces the second point fails.

The theory of regularity structures (RS) gives a framework to solve thisproblem.


The Solution Map on models

Martin’s theory givesI a space of Models (M , d)I a canonical lift of every smooth ⇠" to a model X" 2 MI a continuous function � : M ! D0(Rd) such that u" = �(X")

solves the regularised equation

@tu" = �u" + F(u", ru", ⇠").

The model X" 2 M contains a finite number of relevant explicitproducts. For instance

⇠"(G ⇤ ⇠")(with G the heat kernel). These products can be ill-defined in the limit" ! 0:

E[⇠"(G ⇤ ⇠")] = %" ⇤ G ⇤ %"(0) ! G(0) = +1.

Therefore in general X" does not converge in (M , d) as " ! 0.Lorenzo Zambotti Technion, December 2017

Renormalised products

The theory identifies a class of equations, called subcritical, for whichit is enough to modify a finite number of products in order to obtain aconvergent lift X" 2 M of ⇠". For instance

⇠"(G ⇤ ⇠") ! ⇠"(G ⇤ ⇠") � E[⇠"(G ⇤ ⇠")].

The model X" 2 M contains all these modified (renormalised)products.

Convergence in (M , d) means (simplifying a lot) convergence of allthese objects as distributions.

Then we define the renormalised solution by u" := �(X").


An imageI��

⇠ ⇠"

X X"

X"

u u"u"

�

D0(Rd) D0(Rd)

M

Figure �: In this figure we show the factorisation of the map ⇠" 7! u" into⇠" 7! X" 7! �(X") = u". We also see that in the space of models M wehave several possible lifts of ⇠" 2 S0(Rd), e.g. the canonical model X" and therenormalised model X"; it is the latter that converges to a model X, thus providing alift of ⇠. Note that u" = �(X") and u = �(X).

satisfying some natural bounds on its cumulants can be successfully renormalisedby means of the following scheme:

• Algebraic step: Construction of the space of models (M , d) and renormalisationof the canonical model M 3 X" 7! X" 2 M , this article.

• Analytic step: Continuity of the solution map � : M ! D0(Rd), [Hai��].• Probabilistic step: Convergence in probability of the renormalised model X" to

X in (M , d), [CH��].• Second algebraic step: Identification of �(X") with the classical solution map

for an equation with local counterterms, [BCCH��].We stress that this procedure works for very general noises, far beyond the Gaussiancase.

�.� Overview of resultsWe now describe in more detail the main results of this paper. Let us start fromthe notion of a subcritical rule. A rule, introduced in Definition �.� below, isa formalisation of the notion of a “class of systems of stochastic PDEs”. Moreprecisely, given any system of equations of the type (�.�), there is a natural way ofassigning to it a rule (see Section �.� for an example), which keeps track of whichmonomials (of the solution, its derivatives, and the driving noise) appear on the right

noise. For example, a canonical solution theory for SDEs driven by fractional Brownian motion canonly be given for H > 1

4 , even though these are subcritical for every H > 0. See in particular theassumptions of [CH��, Thm �.��].


The general procedure

One can summarize the procedure into three steps:I Analytic step Construction of the space of models (M , d) and

continuity of the solution map � : M ! D0(Rd),I Algebraic step Renormalisation of the canonical model

X" ! X" 2 M ,I Probabilistic step Convergence in probability of the renormalised

model X" to X in (M , d),I Combinatorial step Computation of the equation solved by u".

We obtain a renormalised solution u := �(X), also the unique solutionof a fixed point problem.

This works for very general noises, far beyond the Gaussian case.


Wong-Zakai for SPDEs

The analogous result for the SPDE is much more subtle: if

@tu" = @2x u" + H(u") + F(u") ⇠", x 2 R,

then u" = �(X") does not converge in general; necessary torenormalise the equation and study u" := �(X"):

@tu" = @2x u" + H(u") � C" F0(u") F(u") + F(u") ⇠"

with C" = E[⇠"(G ⇤ ⇠")] ⇠ "�1. The limit u := �(X) solves

du = (@2x u + H(u)) dt + F(u) dWt

in the Itô sense (true for very general ⇠", see [Chandra-Shen]).

Although there is nothing singular in this SPDE, the result is far fromsimple and requires the full power of the theory [Hairer-Pardoux15].


Another example: KPZ

The regularised version is

@tu" = @2x u" + (@xu")

2 + ⇠"

which has to be renormalised to

@tu" = @2x u" + (@xu")

2 � C" + ⇠"

andC" = E

h

(@xG ⇤ ⇠")2i

⇠ 1".

In this case, one of the ill-defined products to be renormalised is

(@xG ⇤ ⇠")2 �! (@xG ⇤ ⇠")2 � E[(@xG ⇤ ⇠")2].


Integration and Multiplication

Let f , g : [0, T] ! R two continuous functions.

What does it mean to define the integralZ T

0fr gr dr

when f , g are not differentiable ?

Important example: g = B with (Bt)t�0 a Brownian motion.

Starting point of the Rough Paths theory (Terry Lyons, MassimilianoGubinelli).

Example of a more general problem: given a distribution (g) and anon-smooth function ( f ), how can we define their product? Namely adistribution f g.


Local approximation

If g is of class C1, then we define

It :=

Z t

0fr gr dr, t 2 [0, T].

Then we have I0 = 0 and for 0 s t T

It � Is � fs(gt � gs) =

Z t

s( fr � fs) gr dr = o(|t � s|).

We write

I0 = 0, It � Is = fs(gt � gs) + Rst, Rst = o(|t � s|).These properties characterise (It)t2[0,T], since if we have I1 and I2 thensetting I12 := I1 � I2

|I12t � I12

s | = o(|t � s|)which implies I12 constant.


Local approximation

We are studying

It :=

Z t

0fr gr dr, t 2 [0, T].

Let us still study the formula

I0 = 0, It � Is = fs(gt � gs) + Rst, Rst = o(|t � s|).

If we compute for 0 s u t T

Rst � Rsu � Rut = ( fu � fs)(gt � gu)

which does not depend on I.

Therefore the existence of I is equivalent to the existence of R such thatthe above formula holds.


A cochain complex

Les us define for n � 1

�n := {(t1, . . . , tn) 2 [0, T]n : t1 · · · tn},

Cn := {f : �n ! R continuous},

�n : Cn ! Cn+1, (�n f )t1...tn+1 =n+1X

k=1

(�1)n+2�k ft1... tk\ ...tn+1 .

Then we haveI �n+1 � �n ⌘ 0 (exercise!)I if g 2 Cn+1 and �n+1 g = 0, then g = �n f with f 2 Cn (exercise!).

In particular we have an exact cochain complex

R ! C1�1�! C2

�2�! C3�3�! · · ·


Local approximation

Therefore, existence of I 2 C1 such thatI I0 = 0,I (�1 I)st = fs(gt � gs) + o(|t � s|), where (�1 I)st = It � Is,

is equivalent to the existence of R 2 C2 such thatI (�2 R)sut = ( fu � fs)(gt � gu), where (�2 R)sut = Rst � Rsu � Rut,I Rst = o(|t � s|).

Gubinelli calls I the integral, Ast := fs(gt � gs) the germ, and Rst theremainder.


The sewing lemma

For � > 0 and h 2 Cn we set

khk� := sup(t1,...,tn)2�n

|h(t1, . . . , tn)||tn � t1|�

and we say that h 2 C�n if khk� < +1. We also set C�+

n := [�>�C�n .

Theorem (Gubinelli)There exists a unique map ⇤ : C1+

3 \ �2C2 ! C1+2 such that

�2⇤ = idC1+3 \ �2C2

. Moreover ⇤ satifies for all � > 1

k⇤Bk� K�kBk� , B 2 C1+3 \ �2C2.Proof.

See the first lecture sheet of MG


https://www.iam.uni-bonn.de/abteilung-gubinelli/teaching/rough-paths-controlled-paths-ws1516/

Riemann sums

By the Sewing Lemma for a germ A such that �2A has suitableproperties, we can write

Ast = It � Is � Rst, |Rst| C|t � s|1+

and therefore, if P is a partition of [0, t] with |P| going to zero,X

ti2PAtiti+1 = It � I0 �

X

ti2PRtiti+1

andX

ti2P

�

�Rtiti+1

�

� t|P| ! 0

as |P| ! 0. Therefore

It = lim|P|!0

X

ti2PAtiti+1 .


Dyadic approximation

Let us consider for tni := i2�n and n � 0

Int =

X

i�1(tni t) Atni�1tni .

Then, since tn+12i = tn

i , for t 1

|Int � In+1

t | =

�

�

�

�

�

�

X

i�1(tni t)

⇣

Atni�1tni � Atn+12i�2tn+1

2i�1� Atn+1

2i�1tn+12i

⌘

�

�

�

�

�

�

2n

X

i=1

�

�

�

(�2A)tn+12i�2tn+1

2i�1tn+12i

�

�

�

. 2�n(�+��1)

which is summable. Then we obtain that Int ! It as n ! +1 (see

again MG )


https://www.iam.uni-bonn.de/abteilung-gubinelli/teaching/rough-paths-controlled-paths-ws1516/

A first application: Young integration

TheoremIf f 2 C� , g 2 C� (standard Hölder spaces) with � + � > 1 then thereexists a unique pair (I, R) 2 C� ⇥ C�+�

2 such that

I0 = 0, It � Is = fs(gt � gs) + Rst.

The mapC� ⇥ C� 3 ( f , g) ! I 2 C�

is the unique continuous extension of

C1 ⇥ C1 3 ( f , g) !Z •

0f g du 2 C1.


Proof

I Existence. Setting Ast := fs(gt � gs) 2 C�2 , we already know that

(�2A)sut = �( fu � fs)(gt � gu), 0 s t T , so that

|(�2A)sut| C |u � s|� |t � u|� C |t � s|�+� .

Setting R := �⇤�2A 2 C�+�2 then A + R 2 C�

2 and�2(A + R) = �2A � �2⇤�2A = 0, so that A + R = �1I with I 2 C� .

I Uniqueness. If I1, I2 then |I12t � I12

s | = o(|t � s|).I Continuity. The estimate

kIkC� . k f kC�kgkC�

follows from

k⇤�2Ak�+� K�+�k�2Ak�+� , �2A 2 C�+�3 \ �2C2.

in the Sewing Lemma.


If � = � > 1/2

TheoremIf f , g 2 C� , with � > 1/2 then there exists a unique pair(I, R) 2 C� ⇥ C2�

2 such that

I0 = 0, It � Is = fs(gt � gs) + Rst.

In the above situation, we write

It =: I[0,t]( f , g) =:

Z t

0f dg.

Then uniqueness yields the Integration by parts formula

I[0,t]( f , g) + I[0,t](g, f ) = ftgt � f0g0,

since

ftgt � fsgs| {z }

It�Is

= fs(gt � gs) + gs( ft � fs)| {z }

Ast

+ ( ft � fs)(gt � gs)| {z }

Rst

.


If � = � 1/2

However, if � = � 1/2 then neither existence nor uniqueness.

This problem is revelant for stochastic integration and SDEs:

Xt = X0 +

Z t

0�(Xs) dBs

with (Bt)t�0 a standard Brownian motion.

In particular, we can not apply the Sewing Lemma to the germAst := fs(gt � gs) since 2� 1 and therefore in general �2A /2 C1+

3 .

We need to change the germ A in such a way that �2A 2 C1+3 .


Modifying the germ

Note that the result of the integration map is supposed to satisfy

It � Is = fs(gt � gs) + Rst, R 2 C2�2 .

Then we could assume that also f satisfies

ft � fs = f 0s (gt � gs) + R0

st, R0 2 C2�2 .

If Y 2 C2�2 is such that (�2Y)sut = (gu � gs)(gt � gu), setting

Ast := fs(gt � gs) + f 0s Yst,

if also f 0 2 C� then

(�2A)sut = � ( fu � fs � f 0s (gu � gs))

| {z }

R0su

(gt � gu) + (f 0s � f 0

u)Yut 2 C3�3 .

If 1/3 < � 1/2 we are in the setting of the Sewing Lemma.Lorenzo Zambotti Technion, December 2017

Rough paths

For g 2 C� , we want Y 2 C2 such that (�2Y)sut = (gu � gs)(gt � gu).

In fact, for g : [0, T] ! R it is enough to set Yst := 12(gt � gs)2, since

(a + b)2 � a2 � b2 = 2ab.

This is a natural choice, which moreover shows how much all this isrelated to generalised Taylor expansions.

However it is not the only possible choice, nor necessarily the mostdesirable. As we’ll see below, Itô integration is not covered by thissetting.

In fact, for any such Y we can set Y 0 := Y + �1h and Y 0 still has thedesired property.

Note that Yst = 12(gt � gs)2 belongs to C2�

2 . For reasons which will beclear later, we require this property for all Y .


Rough and controlled paths

Let us summarise: given � 2 ]1/3, 1/2] and g 2 C� , we call a pair(g, Y) 2 C� ⇥ C2�

2 a Rough Path if

(�2Y)sut = (gu � gs)(gt � gu), 0 s u t T.

A pair ( f , f 0) 2 C� ⇥ C� is controlled by g if

| ft � fs � f 0s (gt � gs)| . |t � s|2� .

We denote by D2�g the space of paths controlled by g.


Integration of controlled paths

In this setting, we can apply the Sewing Lemma to the germAst := fs(gt � gs) + f 0

s Yst and define the integral I 2 C� such that

�1I = A � ⇤�2A, I0 = 0.

Then the integration map acts (continuously) on controlled paths

D2�g 3 ( f , f 0) 7! (I, f ) 2 D2�

g .


Brownian motion in R

Let us suppose that g ⌘ B, a standard Brownian motion in R. Then forall � < 1/2, a.s. B 2 C� . We fix � 2 ]1/3, 1/2].

We set Yst = 12(Bt � Bs)2. For all � < 1/2, a.s. Y 2 C2�

2 .

A path controlled by B is ( f , f 0) 2 C� ⇥ C� such that

| ft � fs � f 0s (Bt � Bs)| . |t � s|2� , 0 s t T.

For all such ( f , f 0) there exists a unique I 2 C� such that I0 = 0 and

|It � Is � fs(Bt � Bs) � f 0s Yst| . |t � s|3� , 0 s t T.

Moreover

|It � Is � fs(Bt � Bs)| . |t � s|2� , 0 s t T.

If the Stratonovich integralR •

0 fs � dBs is well defined, it is equal to I.


Brownian motion in R

Let us suppose that g ⌘ B, a standard Brownian motion in R. Then forall � < 1/2, a.s. B 2 C� . We fix � 2 ]1/3, 1/2].

We set Yst = 12 [(Bt � Bs)2 � (t � s)]. For all � < 1/2, a.s. Y 2 C2�

2 .

A path controlled by B is ( f , f 0) 2 C� ⇥ C� such that

| ft � fs � f 0s (Bt � Bs)| . |t � s|2� , 0 s t T.

For all such ( f , f 0) there exists a unique I 2 C� such that I0 = 0 and

|It � Is � fs(Bt � Bs) � f 0s Yst| . |t � s|3� , 0 s t T.

Moreover

|It � Is � fs(Bt � Bs)| . |t � s|2� , 0 s t T.

If the Itô integralR •

0 fs dBs is well defined, it is equal to I.


An example

Exercise: find a non-adapted f which satisfies above condition.

For instance, if F : C(R+) ⇥ R ! R is such that

|F(·, 0)| M, F(w, ·) 2 C1,

then fr := F(B, Br) and f 0r = @2F(B, Br) does the job.


Remarks

I In the Young situation (� > 1/2), f and g play symmetric rôles.The integral is a bilinear functional

I If � 1/2, the pair (g, Y) is a non-linear object by the constrainton �2Y .

I In particular, rough paths are non-linear objects. This is wherealgebra gets into the picture.

I On the other hand, for a fixed rough path, controlled paths form alinear space and the integral is a linear operator.

I We interpret Yst =:R t

s (gr � gs) dgr. Then

(�2Y)sut =

Z t

s(gr � gs) dgr �

Z u

s(gr � gs) dgr �

Z t

u(gr � gu) dgr

=

Z t

u(gu � gs) dgr = (gu � gs)(gt � gu).


Remarks

I the analytical bound in the Sewing Lemma implies that theintegral is continuous w.r.t. ( f , g, Y).

I This implies that solutions to a Rough Differential Equation arecontinuous w.r.t. the underlying rough path.

I This was the motivation of Terry Lyons when he introducedRough Paths in the first place, and it is called the Continuity of theItô-Lyons map.

I In the classical theory of stochastic calculus and SDEs, one has ingeneral only measurability of the Itô map.

I If we fix the value of Yst =:R t

s (Br � Bs) dBr, then we canintegrate pathwise any path f controlled by B, for instance any'(Br) with Lipschitz '.

I By derivation, the Sewing Lemma gives a well defineddistribution fs dgs.


Lower regularity

If we want to consider a path g : [0, T] ! R with even lower regularity,say g 2 C� with � 1/3, then we have to modify further the germ.

We assume that ( f , f 0, f 00) 2 (C�)3 satisfies

ft � fs = f 0s (gt � gs) + f 00

s(gt � gs)2

2+ Rst, R 2 C3�

2 ,

f 0t � f 0

s = f 00s (gt � gs) + R0

st, R0 2 C2�2 .

Then the germ

Ast := fs(gt � gs) + f 0s(gt � gs)2

2+ f 00

s(gt � gs)3

3!

satisfies

(�2A)sut = �Rsu(gt � gu) � R0su

(gt � gu)2

2� (f 00

s � f 00u )

(gt � gu)3

3!.

If 1/4 < � 1/3 we are in the setting of the Sewing Lemma.Lorenzo Zambotti Technion, December 2017

Compact notations

Let � 2 ]0, 1[ and g 2 C� .

We set Xnst := 1

n!(gt � gs)n, s, t 2 [0, T], n � 0. By Newton’s binomialtheorem

Xnst =

nX

k=0

Xksu Xn�k

ut , s, u, t 2 [0, T]

(a convolution product...). Note that Xn 2 Cn�2 and

(�2Xn)sut =n�1X

k=1

Xksu Xn�k

ut , s, u, t 2 [0, T].

Now we define N = b1/�c, the largest integer such that N� 1.

We say that Z : [0, T] ! R{0,...,N�1} is controlled by X if

Znt =

N�1X

k=n

Zks Xk�n

st + Rnst, n 2 {0, . . . , N � 1}, Rn 2 C(N�n)�

2 .


Compact notations

Then the germ Ast :=N�1X

k=0

Zks Xk+1

st satisfies

(�2A)sut =N�1X

k=0

⇥

Zks (Xk+1

st � Xk+1su ) � Zk

u Xk+1ut

⇤

=N�1X

k=0

Zks

k+1X

i=1

Xk+1�isu Xi

ut �N�1X

k=0

Zku Xk+1

ut

=N�1X

i=0

Xi+1ut

N�1X

k=i

Zks Xk�i

su �N�1X

i=0

Ziu Xi+1

ut

=N�1X

i=0

Xi+1ut

⇥

Ziu � Ri

su⇤ �

N�1X

i=0

Ziu Xi+1

ut

= �N�1X

i=0

Risu Xi+1

ut 2N�1X

i=0

C(N�i+i+1)�3 ⇢ C1+

3 .


Compact notations

We define as above I by I0 = 0 and

�1I = A � ⇤�2A, R := �⇤�2A.

If we set Z : [0, T] ! R{0,...,N�1} by

Z0t = It, Zn

t := Zn�1t , n 2 {1, . . . , N � 1},

then Z is a controlled path (exercise). In other words we have a linearintegration operator

(Z0, . . . , ZN�1) 7! (I, Z0, . . . , ZN�2)

acting on paths controlled by X. This is the generalisation of⇣

f , f (1), . . . , f (N�1)⌘

7!✓

Z ·

0f , f , f (1), . . . , f (N�2)

◆

,

where f (i) := dif /dxi.Lorenzo Zambotti Technion, December 2017

Stratonovich versus Itô

The above setting works well for Stratonovich integration. Indeed, ifg = B is the trajectory of a Brownian motion, then we can setXStrat

st (n) := 1n!(Bt � Bs)n, s, t 2 [0, T], n � 0.

However this is a very special situation. Why? In the Stratonovich casewe have the integration by parts formula

XStratst (n) =

1n!

(Bt � Bs)n =

Z t

sXStrat

su (n � 1) � dBu

so that two operations, a product and an integration, give the sameresult.This is not true for Itô, since

12(Bt � Bs)

2 =

Z t

s(Bu � Bs) dBu � (t � s).

The two operations of multiplication and integration have to be clearlydistinguished.


Branched rough paths

The idea, again due to Massimiliano Gubinelli, is to use (non-planar)rooted trees to code the relevant integrals. For instance

⌧ =

I XItost (⌧) = 1

I XItost (⌧) = Bt � Bs =

Z t

sdBu

I XItost (⌧) = (Bt � Bs)

3

I XItost (⌧) =

Z t

s(Bu � Bs) dBu

I XItost (⌧) =

Z t

s(Bu � Bs)

2 dBu


The vector case

If B = (B1, . . . , Bd) is d-dimensional BM, we consider rooted treeswith edges decorated by {1, . . . , d}

⌧ =i i j k

j

i

j k

i

I XItost (⌧) = 1

I XItost (⌧) = Bi

t � Bis =

Z t

sdBi

u

I XItost (⌧) = (Bi

t � Bis)(B

jt � B j

s)(Bkt � Bk

s)

I XItost (⌧) =

Z t

s(B j

u � B js) dBi

u

I XItost (⌧) =

Z t

s(B j

u � B js)(B

ku � Bk

s) dBiu


The Connes-Kremier Hopf algebra

We call H the vector space generated by such decorated rooted trees.We endow H with two operations:

I a multiplication or product M : H ⌦ H ! H, linearly generatedby

M(⌧ ⌦ �) := (⌧ t �)/{%⌧ = %�} =: ⌧�

where %⌧ is the root of ⌧ .I a comultiplication or coproduct � : H ! H ⌦ H, linearly

generated by�⌧ :=

X

�✓⌧

(⌧/�) ⌦ �

where � varies among all subtrees of ⌧ with the same root as ⌧and ⌧/� is the tree obtained by contracting � to the root in ⌧ .

This is the Connes-Kreimer Hopf algebra, well known inrenormalisation and in geometric numerical analysis.


Examples

⌧ =`

� =

j k

i⌧� =

`

j k

i

�

j k

i=

j k

i⌦ +

j k⌦

i

+k

⌦

j

i+

j⌦

k

i+ ⌦

j k

i


A recursive formula

H has a recursive structure: all elements of H are obtained from • witha finite number of products and of applications of the operators

⌧ ! [⌧ ]i

where we add to the root of ⌧ a new edge with decoration i and wemove the root to the new node.

The coproduct � has the recursive construction

�• = • ⌦ •, �(⌧1 · · · ⌧n) = (�⌧1) · · · (�⌧n)

�[⌧ ]i = [⌧ ]i ⌦ • + (id ⌦ [·]i)�⌧.


Motivation

This allows to define XItost : H ! R recursively as a linear functional:

1. XItost (•) = 1

2. XItost (⌧1 · · · ⌧n) = XIto

st (⌧1) · · · XItost (⌧n)

3.XIto

st [⌧ ]i =

Z t

sXIto

sr (⌧) dBir

In fact this definition holds in the Stratonovich case or for a smooth Xor for a semimartingale Y , if in the third condition dBi

r is replaced by,respectively,

� dBir, Xi

r dr, dYir.


The Chen relation

The analog of the relation

Xnst =

nX

k=0

Xksu Xn�k

ut , s, u, t 2 [0, T] (3)

becomes now

TheoremFor all s u t:

Xst(⌧) = (Xsu ⌦ Xut)�⌧, 8⌧ 2 H.


Proof of the Chen relation

hXst, [⌧ ]ii =

Z t

s(Xsr⌧) dBi

r

=

Z u

s(Xsr⌧) dBi

r +

Z t

u(Xsr⌧) dBi

r

= hXsu, [⌧ ]ii +

Z t

uhXsu ⌦ Xur, �⌧i dBi

r

= hXsu, [⌧ ]ii + hXsu ⌦Z t

uXur dBi

r, �⌧i= hXsu, [⌧ ]ii + hXsu ⌦ Xut[·]i, �⌧i= hXsu ⌦ Xut, [⌧ ]i ⌦ 1 + (id ⌦ [·]i)�⌧i= hXsu ⌦ Xut, �[⌧ ]ii.

This also holds for the other possible integrals:dBi

r ! � dBir, Xi

r dr, dYir.


Branched rough paths

In 2006 Massimiliano defines a branched rough path of regularity� > 0 as a function X : [0, T]2 ! H⇤ s.t.

I hXst, ⌧i = hXsu ⌦ Xut, �⌧i, 8⌧ 2 H.I hXst, ⌧1⌧2i = hXst, ⌧1ihXst, ⌧2i.I sups 6=t[|hXst, ⌧i|/|t � s|�|⌧ |] < +1, for all ⌧ 2 H, where

|⌧ | := #{edges of ⌧}.

Notations and presentation follow [Hairer-Kelly 2013].

The second condition means that Xst is a character over the algebra H.

The first condition is a lift to H⇤ of the concatenation of paths:

(Xr)r2[s,t] = (Xr)r2[s,u] ? (Xr)r2[u,t].


The extension Theorem

Theorem (T. Lyons, M. Gubinelli)Given a branched rough path X of regularity � > 0, the values(X⌧, |⌧ | = m, m > N) are uniquely determined by the values of(X⌧, |⌧ | = m, m N), where N := b1/�c.

Proof.By recurrence on m � N + 1. We have for all ⌧ such that |⌧ | = m

(�2X⌧)sut = (Xsu ⌦ Xut)�0⌧

where �0⌧ := �⌧ � • ⌦ ⌧ � ⌧ ⌦ • is the reduced coproduct. Weconclude by recurrence and by the Sewing Lemma since we obtain(�2X⌧) 2 Cm�

3 .As I said at the beginning of the course, you need control only finitelymany objects.


Controlled Paths

Given a branched rough path X of regularity � > 0, we say thatZ : [0, T] ! Span{⌧ : |⌧ | N � 1}, with N := b1/�c, is a controlledpath if for all trees ⌧ with |⌧ | N � 1

Z⌧t =

X

|�|N�1

Z�s (Xst ⌦ ⌧⇤)�� + R⌧

st, R⌧ 2 C(N�|⌧ |)�2 ,

where ⌧⇤ : H ! R is the linear functional such that ⌧⇤(�) = (⌧=�)

and |�| is the number of edges in �.

This is the analog of the condition: Z : [0, T] ! R{0,...,N�1} iscontrolled by X if

Znt =

N�1X

k=n

Zks Xk�n

st + Rnst, n 2 {0, . . . , N � 1}, Rn 2 C(N�n)�

2 .


Rough IntegrationTheoremIf Z is a controlled path by X we set for i 2 {1, . . . , d} the germ

Aist :=

X

|�|N�1

Z�s X[�]i

st .

Then there exists a unique function I : R+ ! R such that I0 = 0 and�

�It � Is � Aist

�

� C|t � s|(N+1)� , t, s � 0.

We call I the rough integral of Z against Xi,

It =:

Z t

0Z dXi, Xi

t � Xis := Xst([•]i).

Finally Z is a new controlled path, where

Z⌧t = It (⌧=•) +

X

|�|N�2(⌧=[�]i) Z�

t .


Proof

We prove that �2Ai 2 C(N+1)�3 and then we apply the Sewing Lemma.

(�2Ai)sut =X

|�|N�1

n

Z�s

⇣

X[�]ist � X[�]i

su

⌘

� Z�u X[�]i

ut

o

=X

|�|N�1

n

Z�s

⇣

(Xsu ⌦ Xut)�[�]i � X[�]isu

⌘

� Z�u X[�]i

ut

o

=X

|�|N�1

n

Z�s (Xsu ⌦ Xut)(id ⌦ [·]i)�� Z�

u X[�]iut

o

=X

|⌧ |N�1

8

<

:

X[⌧ ]iut

X

|�|N�1

Z�s (Xsu ⌦ ⌧⇤)�� Z⌧

uX[⌧ ]iut

9

=

;

= �X

|⌧ |N�1

R⌧suX

[⌧ ]iut 2 C(N+1)�

3 .


Composition rule

If ' : R ! R is C1 and Z is a controlled path, then we set forZt := Z•

t['(Zt)]

• = '(Zt),

['(Zt)]⌧ =

N�1X

n=1

X

⌧1···⌧n=⌧

1n!'(n)(Zt) Z⌧1

t · · · Z⌧nt .

Then this is a path controlled by X.


Rough Differential Equations

For a smooth coefficient � : R ! R we want to study the equation

Zt = Z0 +

Z t

0�(Zr) dXr

in the rough integral sense, namely, whereZ : [0, T] ! Span{⌧ : |⌧ | N � 1} is a controlled path.

One defines a norm on controlled paths

kZk =X

|⌧ |N�1

kR⌧k(N�|⌧ |)�

and then uses a fixed point argument.


Convolution product and characters

We can now discuss an important property of the coproduct� : H ! H ⌦ H, coassociativity:

(� ⌦ id)� = (id ⌦ �)�.

We define the convolution product of two linear functionals a, b 2 H⇤

a ? b 2 H⇤, (a ? b)(⌧) := (a ⌦ b)�⌧, ⌧ 2 H.

Then coassociativity of � is equivalent to associativity of ?

(a ? b) ? c = a ? (b ? c).

On H, the set of characters

G := {a 2 H⇤ : a(⌧�) = a(⌧)a(�), 8 ⌧,� 2 H}

forms a group under the convolution product ?.Lorenzo Zambotti Technion, December 2017

A new compact notation

Given a rough path X of regularity � > 0, let us denote by D�X the

space of paths controlled by X.

We consider now the linear operator �st : H ! H

�ts⌧ := (Xst ⌦ id)�⌧ =X

�

Xst(⌧/�)�

where the sum runs over the subtrees � ✓ ⌧ with the same root as ⌧ .

Then a calculation shows that Z 2 D�X if and only if

(Zt � �tsZs)⌧ (=: R⌧

st) 2 C(N�|⌧ |)�2 , 8 |⌧ | N � 1, t, s � 0.

Here and in the following, we write U =P

⌧ U⌧ ⌧ 2 H, namely if ⌧ isa tree then U⌧ is the coefficient multiplying ⌧ in U.


Chen relation revisited

Therefore the Chen relation

Xst = Xsu ? Xut = (Xsu ⌦ Xut)�

becomes�ts = �tu � �us.

These operators are invertible since

(�ts)�1 = (X�1

ts ⌦ id)� = (Xst ⌦ id)�.

where the first inverse is in the sense of linear operators and the secondis in the group G endowed with the product ?.

In fact, the Chen relation says that we can set Xt := X0t and obtain

Xst = X�1s ? Xt.


Local approximations

For a given rough path X and Z 2 D�X we have characterised

R •0 Z dX

as the only function I such that I0 = 0 and�

�

�

�

�

�

It � Is �X

|�|N�1

Z�s X[�]i

st

�

�

�

�

�

�

C|t � s|(N+1)� .

We interpret (Xst⌧, |⌧ | N) as a fixed family of generalised Taylormonomials in t centered at s. Then (Z�

s , |�| N � 1) is family ofcoefficients.

Therefore the problem here is to find a function I which is locally wellapproximated by the above generalised Taylor expansion.

Even in the classical case, such a function I may not exist if thecoefficients of the Taylor expansion are not chosen in a consistent way.The condition which ensures that Z 2 D�

X turns out to be sufficient.Then the Sewing Lemma transforms consistency of Z in existence of I.


Singular stochastic PDEs

Around 2010, Martin and Massimiliano, among others, try togeneralise Rough Paths to stochastic PDEs like KPZ, PAM and �4.

(KPZ) @tu = �u + (ru)2 + ⇠, (t, x) 2 R ⇥ R,

(PAM) @tu = �u + u ⇠, (t, x) 2 R ⇥ R2,

(�43) @tu = �u � u3 + ⇠, (t, x) 2 R ⇥ R3.

This needs two generalisations:I The rough path must be parametrized by Rd with d � 2, but the

Sewing Lemma is very much one-dimensionalI Xst(⌧) can become a distribution, say, in t for fixed s, i.e. we want

to allow that sups 6=t[|Xst(⌧)|/|t � s|�⌧ ] < +1 with �⌧ 2 R.Two new theories are born: regularity structures and paraproducts.


Regularity Structures

DefinitionA regularity structure T = (A, T, G) consists of the followingelements:

I An index set A ⇢ R such that A is bounded from below, and A islocally finite.

I A model space T , which is a graded vector space T =L

↵2A T↵,with each T↵ a Banach space.

I A structure group G of linear operators acting on T such that, forevery � 2 G, every ↵ 2 A, and every a 2 T↵, one has

�a � a 2M

�<↵

T� .

We say that T↵ is a subspace of homogeneity or degree ↵.


Example: Rough Paths

Given � > 0 and a dimension d 2 N, we setI A := �NI T := H = Span{rooted trees decorated by {1, . . . , d}},

Tn� := Span{⌧ : |⌧ | = n}I G is the set of all linear g : H ! R such that g(⌧�) = g(⌧)g(�)

for all ⌧,� 2 H (characters), and for all g 2 G we set

�g : H ! H, �g⌧ := (g ⌦ id)�⌧ =X

�

g(⌧/�)�.

Note that �ts = �Xst .

In the general case, we allow A to have negative elements, in order torepresent distributions.


Models

A model of T = (A, T, G) is a pair (⇧x, �xz)x,z2Rd of operators s.t.I ⇧x : T ! D0(Rd) is linear for all x 2 Rd

I �xz : T ! T is an element of G for all x, z 2 Rd

I for all x, y, z 2 Rd

�yx � �xz = �yz, ⇧x�xz⌧(y) = ⇧z⌧(y), ⌧ 2 T.

I (⇧x, �xz)x,z2Rd satisfy several analytical requirements.

The distributions ⇧x(y) are our generalised monomials in the variable yand centered at x.

The operators �xz will be used to compare coefficients at x and z andcheck a consistency relation as for controlled paths of rough paths.


Comparison with the Chen relation

Let us compare the algebraic requirement

�yx � �xz = �yz, ⇧x�xz⌧(y) = ⇧z⌧(y), ⌧ 2 T.

with the Chen relation for Rough Paths

Xzx ? Xxy = Xzy, �yx � �xz = �yz.

The above formulae are an astute generalisation, where we multiply �by �, ⇧ by � but never ⇧ by ⇧, since in general ⇧x : T ! D0(Rd).


Example: Rough Paths as Models

For � > 0 and d 2 N fixed, a rough path X of regularity � gives amodel of the above regularity structure if we set

I ⇧x⌧(y) := Xxy⌧

I �xz⌧ := (Xzx ⌦ id)�⌧ .

Then by the Chen relation

⇧x�xz⌧(y) = (Xzx ⌦ Xxy)�⌧ = Xzx⌧ = ⇧z⌧(y).

The analytical requirements for a Rough Path and a Model coincide inthis case.


The analog of Controlled Paths

Given a regularity structure T = (A, T, G) and a model (⇧x, �xz), wedefine for all � 2 R

D� :=

8

<

:

U : Rd !M

�<�

T� : |(Ux � �xzUz)⌧ | C|x � z|�� , 8 ⌧ 2 T�

9

=

;

This is called the space of modelled distributions.

Again, this condition must be interpreted as a consistency relationbetween coefficients of a generalised Taylor expansion.

This is the space where one can lift an equation (SPDE) as for RoughDifferential Equations and find a solution by means of a fixed pointargument.


The Reconstruction Theorem

The main result of the theory is the following

Theorem (Hairer)

I If � > 0 then there exists a unique (linear) mapR : D� ! D0(Rd) such that (informally) for all U 2 D�

|RU(y) � ⇧xU(x)(y)| C|y � x|� .

I If � 0 then there exists a (in general non-unique) linear mapR : D� ! D0(Rd) such that (informally) for all U 2 D�

|RU(y) � ⇧xU(x)(y)| C|y � x|� .

We call R the reconstruction operator. As the Sewing Lemma forcontrolled paths, given a consistent family of coefficients it(re)constructs a distribution which is locally approximated by theassociated Taylor expansion.


Models for SPDEs

Consider e.g.

@tu" = �u" + �(u") ⇠", (t, x) 2 R ⇥ R.

What is the associated "Rough Path" (model) ? If we had before

hXst, [⌧ ]ii =

Z t

s(Xsu⌧) Xi

u du

then now it looks reasonable to replace

Xiu �! ⇠"(u, y),

Z t

s· · · du �!

Z t

0

Z

RGt�u(x � y) · · · du dy.


Rough Paths ?

In Rough Paths Xst(⌧) is always an increment

hXst, [⌧ ]ii =

Z t

s(Xsu⌧) Xi

u du

=

Z t

a(Xsu⌧) Xi

u du �Z s

a(Xsu⌧) Xi

u du.

The analytic property

sups6=t

[|hXst, ⌧i|/|t � s|�|⌧ |] < +1

is recursive, since if s, t are close to each other then u 2 [s, t] is close tos as well.


Trees

Trees can be built as followsI 1, Xi, ⌅ are trees, i = 1, . . . , dI the product of trees (by identifying the roots) is a treeI if ⌧ is a tree then Ik(⌧) is a tree, with k 2 Nd.

Let us denote by H the vector space generated by all such trees.

Examples: I(⌅), Xn⌅Ik(⌅), I((Ik(⌅))2)

In Branched Rough Paths (BRP), trees are generated byI a single root 1I a product consisting of identification of rootsI integration operators ⌧ ! [⌧ ]i which add trunks at the root.

In BRP we do not have trees like ⌅, Xi.


Homogeneity

Let s = (2, 1, . . . , 1) 2 Nd. For k 2 Nd we set

|k|s = s1k1 + · · · + sdkd.

To a tree ⌧ we associate a real number |⌧ |s called its homogeneity:|⌅|s = ↵ < 0, |Xi|s = si, |1|s = 0

|⌧1 · · · ⌧n|s = |⌧1|s + · · · + |⌧n|s, |Ik(⌧)|s = |⌧ |s + 2 � |k|s.

In the following we consider the case ↵ = �3/2 � � for � > 0 small,relevant for space-time white noise in one space-dimension, i.e. d = 2.

In BRP the homogenity of a tree is equal to �|⌧ | where � > 0 is theregularity parameter and |⌧ | is the number of edges. In particular it isalways � 0.


Models

Let us use a now notation for the addition of a new trunk:

[⌧ ]i �! I(⌧).

For SPDEs, we imagine a recursive object ⇧x⌧(y) replacing Xst(⌧),such that

⇧x I(⌧)(y) = G ⇤ (⇧x⌧)(y) � G ⇤ (⇧x⌧)(x).

(From now on, x, y are space-time variables.)What would be a reasonable analytic requirement here ? If

|⇧x⌧(y)| C|y � x||⌧ |s

with |⌧ |s > 0 then we would like to have, by analogy with RPs,

|⇧x I(⌧)(y)| C|y � x||⌧ |s+2

but this requires further assumptions on y 7! G ⇤ (⇧x⌧)(y).Lorenzo Zambotti Technion, December 2017

Taylor sums and remainders

In fact we have to modify the definition of ⇧x⌧(y). We recall

hXst, [⌧ ]ii =

Z t

s(Xsu⌧) Xi

u du

=

Z t

a(Xsu⌧) Xi

u du �Z s

a(Xsu⌧) Xi

u du.

This increment is a Taylor remainder at order 0. This suggests to go toa higher order by setting

⇧x I(⌧)(y) = G ⇤ (⇧x⌧)(y) �X

k<|I(⌧)|s

(y � x)k

k!@kG ⇤ (⇧x⌧)(x).


Tree representation: Examples

⌅ �!Z

'(z) ⇠"(z) dz =

Z

'(z) %"(z � x) ⇠(dx) dz �!z

x

I(⌅) �!Z

'(z) G ⇤ ⇠"(z) dz �!z

x y

⌅I(⌅) �!Z

'(z) ⇠"(z) G ⇤ ⇠"(z) dz �!z

x y2

y1


Tree representation: Examples

⌅I(⌅I(⌅)) ⌅I(⌅)2


Further decorations on trees

We have additional decorations on trees, needed to code

⇧x Ik(⌧)(y) = @kG ⇤ (⇧x⌧)(y) �X

`<|Ik(⌧)|s

(y � x)`

`!@k+`G ⇤ (⇧x⌧)(x).

I n on nodes, representing powers of (y � x)I e on edges, representing derivatives @kG of the heat kernel

We use the notation Tne for such a decorated tree.


Distributions

We have a linear space H of decorated trees, representing distributionson Rd which are relevant to the given equation.

Since we do not expect to multiply all distributions, H is not assumedto be an algebra.

We do not expect H to have a coproduct either, so it is not clear how todefine the Chen relation

Xxz ? Xzy = Xxy.

The solution is to split Xxy into two components, containingrespectively functions (�yz) and distributions (⇧x⌧(z)).

This is done with comodules.


Remarks

I in general ⇧x : H ! D0(Rd) is not multiplicative, even if it takesvalues in smooth functions

I this "freedom" of ⇧x to be non-multiplicative is crucial in therenormalisation procedure

I the canonical choice of ⇧x, for a regularised version ⇠" of thenoise, satisfies moreover multiplicativity

⇧x(⌧1 · · · ⌧n) = ⇧x(⌧1) · · · ⇧x(⌧n).

The renormalisation procedure will destroy multiplicativity, as in

⇠"(G ⇤ ⇠") ! ⇠"(G ⇤ ⇠") � E[⇠"(G ⇤ ⇠")].


The canonical ⇧x operators

We fix a smooth bounded realization of the noise ⇠. We define acanonical smooth model recursively

⇧x1(y) = 1, ⇧xX(y) = (y � x), ⇧x⌅(y) = ⇠(y),

⇧x(⌧1 · · · ⌧n)(y) =n

Y

j=1

⇧x⌧j(y),

⇧xIk(⌧)(y) = (G(k) ⇤ ⇧x⌧)(y) �X

|`|s<|Ik(⌧)|s

(y � x)`

`!(G(k+`) ⇤ ⇧x⌧)(x).

Multiplicativity is used in order to define ⇧x, but never afterwards.

In BRP we have Xst, with s $ x and t $ y.


Examples

The for ⌧ = I(⌅), the homogeneity is |I(⌅)|s = 1/2 � � and

⇧xI(⌅)(y) = (G ⇤ ⇠)(y) � (G ⇤ ⇠)(x)

For ⌧ = I(⌅I(⌅)) the homogeneity is |I(⌅I(⌅))|s = 1 � 2� and

⇧x⌧(y) = G ⇤ (⇠⇧xI(⌅))(y) � G ⇤ (⇠⇧xI(⌅))(x) =

= G ⇤ (⇠(G ⇤ ⇠ � G ⇤ ⇠(x)))(y) � G ⇤ (⇠(G ⇤ ⇠ � G ⇤ ⇠(x)))(x).

For ⌧ = I(⌅I(⌅I(⌅))), the homogeneity is 3/2 � 3� > 1 and

⇧x⌧(y) = G ⇤ (⇠⇧xI(⌅I(⌅)))(y) � G ⇤ (⇠⇧xI(⌅I(⌅)))(x)

� (y � x)G0 ⇤ (⇠⇧xI(⌅I(⌅)))(x).


The canonical � operators

We can use the recursive definition (due to Yvain Bruned)

�xyX = X + (x � y), �xy⌅ = ⌅, �xy

Y

i

⌧i =Y

i

�xy⌧i

�xyIk(⌧) = Ik(�xy⌧) �X

|`|s<|Ik(⌧)|s

(⇧xIk+`(�xy⌧))(y)(X + x � y)`

`!

One can check by recurrence the compatibility conditions

�yx � �xz = �yz, ⇧x�xz = ⇧z.


Examples

For instance

�xzI(⌅) = I(⌅) + (G ⇤ ⇠)(x) � (G ⇤ ⇠)(z)

Another example:

�xzI(⌅I(⌅)) = (�G ⇤ ⇠(x)G ⇤ ⇠(z) + G ⇤ (⇠G ⇤ ⇠)(x) + G ⇤ ⇠(z)2

� G ⇤ (⇠G ⇤ ⇠)(z))+ (G ⇤ ⇠(x) � G ⇤ ⇠(z)) I(⌅)

+ I(⌅I(⌅))


Examples

And another:

�xzI(⌅I(⌅I(⌅))) = I(⌅I(⌅I(⌅))) + G ⇤ ⇠(x)G ⇤ ⇠(z)2

� G ⇤ ⇠(x)G ⇤ (⇠G ⇤ ⇠)(z) � G ⇤ (⇠G ⇤ ⇠)(x)G ⇤ ⇠(z)+ G ⇤ (⇠G ⇤ (⇠G ⇤ ⇠))(x) + (z � x)(G0 ⇤ (⇠)(z)G ⇤ ⇠(z)2

� G0 ⇤ (⇠)(z)G ⇤ (⇠G ⇤ ⇠)(z) � G0 ⇤ (⇠G ⇤ ⇠)(z)G ⇤ ⇠(z)+ G0 ⇤ (⇠G ⇤ (⇠G ⇤ ⇠))(z)) � G ⇤ ⇠(z)3 + 2G ⇤ ⇠(z)G ⇤ (⇠G ⇤ ⇠)(z)� G ⇤ (⇠G ⇤ (⇠G ⇤ ⇠))(z))+ I(⌅)(�G ⇤ ⇠(x)G ⇤ ⇠(z) + G ⇤ (⇠G ⇤ ⇠)(x) + G ⇤ ⇠(z)2 � G ⇤ (⇠G ⇤ ⇠)(z))+ I(⌅I(⌅))(G ⇤ ⇠(x) � G ⇤ ⇠(z))+ X(G0 ⇤ (⇠)(x)G ⇤ ⇠(x)2 � G0 ⇤ (⇠)(x)G ⇤ (⇠G ⇤ ⇠)(x) � G0 ⇤ (⇠G ⇤ ⇠)(x)G ⇤ ⇠(x)+ G0 ⇤ (⇠G ⇤ (⇠G ⇤ ⇠))(x) � G0 ⇤ (⇠)(z)G ⇤ ⇠(z)2 + G0 ⇤ (⇠)(z)G ⇤ (⇠G ⇤ ⇠)(z)+ G0 ⇤ (⇠G ⇤ ⇠)(z)G ⇤ ⇠(z) � G0 ⇤ (⇠G ⇤ (⇠G ⇤ ⇠))(z))


The canonical ⇧ operators

We define recursively

⇧1(y) = 1, ⇧X(y) = y, ⇧⌅(y) = ⇠(y),

⇧(⌧1 · · · ⌧n)(y) =n

Y

j=1

⇧⌧j(y),

⇧Ik(⌧)(y) = (G(k) ⇤ ⇧⌧)(y).

Multiplicativity is used in order to define ⇧, but never afterwards.

One can find that �xz = ��1x � �z and then

⇧x = ⇧ � �x


The coefficients of the solution of generalized KPZ

@tu = �u + f (u) (@xu)2 + g(u) ⇠.

We lift this equation to a D� space.

At order � = 1/2

U(x) = u(x)1 + g(u(x))I(⌅)



At order � = 1 (gk = dkg/dxk)

u(x)1 + g(u(x))I(⌅) + g1g(u(x))I(⌅I(⌅)) + g2f (u(x))I((I1(⌅))2)

+ (u1(x) � g(u)(G1 ⇤ ⇠)(x) + g1(u)(G1 ⇤ ⇠)(G ⇤ ⇠)(x)� g1g(u)G1 ⇤ (⇠(G ⇤ ⇠))(x) � g2f (u)G1 ⇤ ((G1 ⇤ ⇠)2)(x))X



At order � = 3/2

u(x)1 + g(u(x))I(⌅) + g1g(u(x))I(⌅I(⌅)) + g2f (u(x))I((I1(⌅))2)

+ (u1(x) � g(u)(G1 ⇤ ⇠)(x) + g1(u)(G1 ⇤ ⇠)(G ⇤ ⇠)(x)� g1g(u)G1 ⇤ (⇠(G ⇤ ⇠))(x) � g2f (u)G1 ⇤ ((G1 ⇤ ⇠)2)(x))X

+ 1/2g2g(u(x))2I(⌅(I(⌅))2) + g21g(u(x))I(⌅I(⌅I(⌅)))

+ g1g2f (u(x))I(⌅I((I1(⌅))2)) + 2g3f1(u(x))I((I1(⌅))2I(⌅))

+ 2g1g2f (u(x))I(I1(⌅)I1(⌅I(⌅))) + 2g3f (u(x))2I(I1(⌅)I1((I1(⌅))2))

+ (u1g1(u(x)) � g1g(u)I1(⌅)(x) + g21g(u)(G1 ⇤ ⇠)(G ⇤ ⇠)(x)

� g21g(u)G1 ⇤ (⇠(G ⇤ ⇠))(x) � g1g2f (u)G1 ⇤ ((G1 ⇤ ⇠)2)(x))I(⌅X)

+ (�2g2f (u)I2(⇠)(x) + 2u1gf (u(x)) + 2g1g2f (u)I2(⇠)(x)(G ⇤ ⇠)(x)� 2g1g2f (u)G2 ⇤ (⇠(G ⇤ ⇠))(x) � 2g3f (u)2G2 ⇤ ((G1 ⇤ ⇠)2)(x))I(I1(⌅))


The trees of generalized KPZ⌅, ⌅I(⌅), (I1(⌅))2, ⌅(I(⌅))2, ⌅I(⌅I(⌅)), ⌅I((I1(⌅))2)

(I1(⌅))2I(⌅), I1(⌅)I1(⌅I(⌅)), I1(⌅)I1((I1(⌅))2), ⌅X

I1(⌅), ⌅(I(⌅))3, ⌅I(⌅)I(⌅I(⌅)), ⌅I(⌅)I((I1(⌅))2)

⌅I(⌅(I(⌅))2), ⌅I(⌅I(⌅I(⌅))), ⌅I(⌅I((I1(⌅))2))

⌅I((I1(⌅))2I(⌅)), ⌅I(I1(⌅)I1(⌅I(⌅))), ⌅I(I1(⌅)I1((I1(⌅))2))

(I1(⌅))2(I(⌅))2, (I1(⌅))2I(⌅I(⌅)), (I1(⌅))2I((I1(⌅))2)

I1(⌅)I1(⌅I(⌅))I(⌅), I1(⌅)I1((I1(⌅))2)I(⌅), I1(⌅)I1(⌅(I(⌅))2)

I1(⌅)I1(⌅I(⌅I(⌅))), I1(⌅)I1(⌅I((I1(⌅))2))

I1(⌅)I1((I1(⌅))2I(⌅)), I1(⌅)I1(I1(⌅)I1(⌅I(⌅)))

I1(⌅)I1(I1(⌅)I1((I1(⌅))2)), I1(⌅I(⌅))I1((I1(⌅))2)

(I1(⌅I(⌅)))2, (I1((I1(⌅))2))2, ⌅I(⌅)X, ⌅I(⌅X)

⌅I(I1(⌅)), (I1(⌅))2X, I1(⌅)I(⌅), I1(⌅)I1(⌅X)

I1(⌅)I1(I1(⌅)), I1(⌅I(⌅)), I1((I1(⌅))2)Lorenzo Zambotti Technion, December 2017

Positive and negative renormalisations

From now on I discuss a selection of the content of [BHZ16].

If we setI A+(T) := {S ✓ T : S subtree with the same root as T}I A�(T) := {S ✓ T : S subforest of T}

then we define

�+Tne =

X

S2A+(T)

X

nS,eS

1eS!

✓

n

nS

◆

(T/S)n�nSe+eS

⌦ SnS+⇡eSe

��Tne =

X

S2A�(T)

X

nS,eS

1eS!

✓

n

nS

◆

(T/S)n�nSe+eS

⌦ SnS+⇡eSe


Positive Renormalisation

The operator which is used to express the Chen relation is

�+Tne =

X

S2A+(T)

X

nS,eS

1eS!

✓

n

nS

◆

(T/S)n�nSe+eS

⌦ SnS+⇡eSe

where A+(T) := {S ✓ T : S subtree with the same root as T}

�! ⌦


Negative Renormalisation

The operator which expresses the renormalisation procedure is

��Tne =

X

S2A�(T)

X

nS,eS

1eS!

✓

n

nS

◆

(T/S)n�nSe+eS

⌦ SnS+⇡eSe

where A�(T) := {S ✓ T : S subforest of T}

�! ⌦


Coassociativity and Cointeraction

(�+ ⌦ id)�+ = (id ⌦ �+)�+

(�� ⌦ id)�� = (id ⌦ ��)��

M(1)(24)(3)(�� ⌦ ��)�+ = (�+ ⌦ id)��

These results are simple combinatorial relations at the level of therooted trees/forests, but the presence of the decorations make theformulae substantially more complex.


The algebra H+

We define H+ as the algebra (for the tree product) generated by thetrees ⌧ 2 H such that

I ⌧ = Ik(�) for � 2 H and k 2 Nd

I |⌧ |s � 0.

For fixed x 2 Rd we define recursively the character gx : H+ ! R

gx(1) = 1, gx(X) = �x, gx(⌧1 · · · ⌧n) =n

Y

j=1

gx(⌧j),

gx(Ik(⌧)) = �X

|`|s<|Ik(⌧)|s

(�x)`

`!(G(k+`) ⇤ ⇧x⌧)(x).

We denote by p+ : H ! H+ the canonical projection.


Positive renormalisation

We see now the connection with the canonical smooth model

TheoremWe have for all ⌧ 2 H

⇧x⌧(y) = h(gx � p+) ⌦ ⇧, �+⌧i(y).

Moreover �xz : H ! H is given by

�xz⌧ = (((gz ? g�1x ) � p+) ⌦ id)�+⌧.


Back to renormalisation

The construction above gets into serious trouble if ⇠ wants to become adistribution, which is our ultimate goal.

Formulae (among others) which certainly do not make sense anymore:

⇧x(⌧1 · · · ⌧n)(y) =n

Y

j=1

⇧x⌧j(y), ⇧(⌧1 · · · ⌧n)(y) =n

Y

j=1

⇧⌧j(y).

Indeed, products can diverge

This motivates (again) renormalisation.



We want to understand what is the class of ⇧0 such that the aboveconstruction can be repeated and yields a model, both algebraically andanalytically.

We consider linear ⇧0 : H ! C(Rd) such that

⇧01 = 1, ⇧0X(y) = y, ⇧0⌅(y) = ⇠(y),

⇧0Ik(⌧)(y) = (G(k) ⇤ ⇧0⌧)(y).

We have given up multiplicativity ! We define on H+

g0x(1) = 1, g0

x(X) = �x, g0x(⌧1 · · · ⌧n) =

nY

j=1

g0x(⌧j),

g0x(Ik(⌧)) = �

X

|`|s<|Ik(⌧)|s

(�x)`

`!(G(k+`) ⇤ ⇧0

x⌧)(x).



Now we define ⇧0x : H ! C(Rd) and �0

xz : H ! H⇧0

x⌧(y) = hg0x ⌦ ⇧0, �+⌧i(y),

with a similar formula for �0xz.

TheoremWith these definitions, we have for all x, y, z 2 Rd

�0yx�

0xz = �0

yz, ⇧0x�

0xz⌧(y) = ⇧0

z⌧(y), ⌧ 2 H.

However, in general it is not clear that the following is satisfied:I �xz⌧ = ⌧ +

P

i ⌧i, with |⌧i|s < |⌧ |s (lower triangular)I several analytic requirements, among which

|⇧x⌧(y)| C⌧ |y � x||⌧ |s , if |⌧ |s > 0.


Negative renormalisation

We define the linear space H� generated by all forests ⌧1 t · · · t ⌧n

with ⌧i 2 H and |⌧i|s < 0, with projection p� : H ! H�. These arethe trees (forests) whose ⇧ can diverge.

We recall the definition

��Tne =

X

S2A�(T)

X

nS,eS

1eS!

✓

n

nS

◆

(T/S)n�nSe+eS

⌦ SnS+⇡eSe

and A�(T) := {S ✓ T : S subforest of T}.

�! ⌦


Renormalised models

We define G� as a space of characters (multiplicative functionals) onthe Hopf algebra of forests H� endowed with the coproduct ��.

TheoremLet ` 2 G� and M` : H 7! H

M`⌧ := (id ⌦ ` � p�)��⌧.

For every model (⇧0x, �

0xz), the pair (⇧0

x � M`, �0xz � M`) is also a model.

In particular, if (⇧x, �xz) is the canonical smooth model, then(⇧x � M`, �xz � M`) is a model.


The BPHZ model

If the noise ⇠ is stationary, there exists a special choice of ` 2 G�: forall tree ⌧ with |⌧ |s < 0

`BPHZ(⌧) = Eh

⇧ � A�(⌧)i

(0)

where A� is a map that we call negative twisted antipode.

TheoremThe character `BPHZ 2 G� is the only one such that

E[⇧ � M`BPHZ(⌧)] = 0

for all tree ⌧ with |⌧ |s < 0.

The BPHZ model converges by [Chandra-Hairer16]: for some deepreason, centering is enough for convergence.


An example in �43

R��

decoration. Using the symbolic notation, it is given by ⌧ = I(⌅)2I(I(⌅)3). Thenwe use the following representation:

I(⌅) = , R↵Iei =a

i , Xi = i , J= , ⌧ = ,

where ei is the ith canonical basis element of Nd and a belongs to {↵, �, �} with↵ = 2I+ 2⌅, � = 2I+ 2⌅ + 1 and � = 5I+ 4⌅. Then we have

��ex = ⌦ 11 + 11 ⌦ + 3 ⌦ �

+ 3i

⌦ �

i

+ ⌦�

+i

⌦�

i+ 3 ⌦

�

�

+ 3i

⌦�

�

i

+ 3i

⌦�

�

i+ 3

i j⌦

�

�

j

i+ ⌦

�+ (...)

with summation over i and j implied. In (...), we omit terms of the form ⌧ (1) ⌦ ⌧ (2)

where ⌧ (1) may contain planted trees or where ⌧ (2) has an edge of type Ifinishing ona leaf. The planted trees will disappear by applying an element of Gex

� and the othersare put to zero through the evaluation of the smooth model � see [Hai��, Ass. �.�]where the kernels {Kt}t2L+ are chosen such that they integrate polynomials to zeroup to a certain fixed order. If g 2 Gex

� is the character associated to the BPHZrenormalisation for a Gaussian driving noise with a covariance that is symmetricunder spatial reflections, we obtain

M exg ⌧ = (g ⌦ id)��

ex⌧

= + 3C1�

+ C1 �+ 3C2

1 �

�

+ 3C2 �

whereC1 = �g�(�)

h i

, C2 = �g�(�)h i

,

and all other renormalisation constants vanish. Applying Q, we indeed recover therenormalisation map given in [Hai��, Sec �.�]. The main interest of the extendeddecorations is to shorten some Taylor expansions which allows us to get the co-interaction between the two renormalisations. In the computation below, we showthe di�erence between a term having extended decoration and the same without:


An example in �43

R��

decoration. Using the symbolic notation, it is given by ⌧ = I(⌅)2I(I(⌅)3). Thenwe use the following representation:

I(⌅) = , R↵Iei =a

i , Xi = i , J= , ⌧ = ,

where ei is the ith canonical basis element of Nd and a belongs to {↵, �, �} with↵ = 2I+ 2⌅, � = 2I+ 2⌅ + 1 and � = 5I+ 4⌅. Then we have

��ex = ⌦ 11 + 11 ⌦ + 3 ⌦ �

+ 3i

⌦ �

i

+ ⌦�

+i

⌦�

i+ 3 ⌦

�

�

+ 3i

⌦�

�

i

+ 3i

⌦�

�

i+ 3

i j⌦

�

�

j

i+ ⌦

�+ (...)

with summation over i and j implied. In (...), we omit terms of the form ⌧ (1) ⌦ ⌧ (2)

where ⌧ (1) may contain planted trees or where ⌧ (2) has an edge of type Ifinishing ona leaf. The planted trees will disappear by applying an element of Gex

� and the othersare put to zero through the evaluation of the smooth model � see [Hai��, Ass. �.�]where the kernels {Kt}t2L+ are chosen such that they integrate polynomials to zeroup to a certain fixed order. If g 2 Gex

� is the character associated to the BPHZrenormalisation for a Gaussian driving noise with a covariance that is symmetricunder spatial reflections, we obtain

M exg ⌧ = (g ⌦ id)��

ex⌧

= + 3C1�

+ C1 �+ 3C2

1 �

�

+ 3C2 �

whereC1 = �g�(�)

h i

, C2 = �g�(�)h i

,

and all other renormalisation constants vanish. Applying Q, we indeed recover therenormalisation map given in [Hai��, Sec �.�]. The main interest of the extendeddecorations is to shorten some Taylor expansions which allows us to get the co-interaction between the two renormalisations. In the computation below, we showthe di�erence between a term having extended decoration and the same without:Lorenzo Zambotti Technion, December 2017

www.rism.it [email protected]

D. Cassani (Chair) - C.D. Pagani - B. Ruf G. Tessitore - F. Tomarelli

SCIENTIFIC COMMITTEE

MAIN COURSES SUPPORTED BY

T H E R I E M A N N I N T E R N AT I O N A L S C H O O L

O F M AT H E M AT I C S

Developments in StochasticDevelopments in StochasticDevelopments in Stochastic Partial Differential EquationsPartial Differential EquationsPartial Differential Equations

in honor of Giuseppe Da Pratoin honor of Giuseppe Da Pratoin honor of Giuseppe Da Prato

G. Da Prato, SNS - Pisa

F. Flandoli, SNS - Pisa

A. Lunardi, Università di Parma

SCIENTIFIC BOARD OF RISM

ORGANIZATION D. Cassani, Università degli Studi dell’Insubria M. Fuhrman, Università degli Studi di Milano G. Guatteri, Politecnico di Milano F. Masiero, Università degli Studi di Milano Bicocca G. Tessitore, Università degli Studi di Milano Bicocca

M. Hairer (Director), Imperial College - London S. Cerrai, University of Maryland A. Debussche, ENS - Rennes L. Zambotti, Université Pierre et Marie Curie - Paris

Varese, July 23 - 27, 2018

E. Bombieri, IAS - Princeton R. Donagi, University of Pennsylvania I. Ekeland, Université de Paris-Dauphine M. Hairer, Imperial College - London L. Nirenberg, CIMS - New York A. Quarteroni, Politecnico di Milano & EPFL - Lausanne

PLENARY LECTURES

A. Debussche, ENS - Rennes

M. Hairer, Imperial College - London

F. Otto, Max Planck Institute - Leipzig

MANAGING BOARD OF RISM

http://www.rism.it

mailto:[email protected]

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