Stochastic regularization effects of semi-martingales on random functions

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Stochastic regularization effects of semi-martingales on random functions

Romain Duboscq, Anthony Réveillac

To cite this version:

Romain Duboscq, Anthony Réveillac. Stochastic regularization effects of semi-martingales on random functions. Journal de Mathématiques Pures et Appliquées, Elsevier, 2016. �hal-01178614v3�

Stochastic regularization effects of semi-martingales on random functions

Romain Duboscq

Anthony Réveillac

INSA de Toulouse

IMT UMR CNRS 5219

Université de Toulouse

Abstract

In this paper we address an open question formulated in [17]. That is, we extend the Itô-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X andF the solution of the Fokker-Planck equation associated to X, to random mappingsf. To this end we provide new results on a class of adpated and non-adapted Fokker-Planck SPDEs and BSPDEs.

Key words: Stochastic regularization; (Backward) Stochastic Partial Differential Equa- tions; Malliavin calculus.

AMS 2010 subject classification: Primary: 35R60; 60H07; Secondary: 35Q84.

1 Introduction

In [17], the authors analyzed the effects of a multiplicative stochastic perturbation on the well-posedness of a linear transport equation. One of the key tool in their analysis is the so-calledItô-Tanaka trick which links the time-average of a functionf depending on a stochastic process and F the solution of the Fokker-Planck equation associated to the stochastic process. More precisely, the formula reads as

Z T 0

f(t, X_t^x)dt=−F(0, x)− Z T

0

∇F(t, X_t^x)·dW_t, P−a.s. (1.1) where (X_t^x)_t≥0 is a solution of the stochastic differential equation

X_t^x=x+ Z t

0

b(s, X_s^x)ds+W_t, (1.2) and F is the solution of the backward Fokker-Planck equation

F(t, x) =− Z T

t

1

2∆ +b(s, x)· ∇

F(s, x)ds− Z T

t

f(s, x)ds. (1.3)

1romain.duboscq@insa-toulouse.fr

2anthony.reveillac@insa-toulouse.fr

3135 avenue de Rangueil 31077 Toulouse Cedex 4 France

In [24], by means of suitable regularity results for solutions of parabolic equations in L^q−L^p spaces, the authors showed, assuming f, b ∈ E := L^q([0, T];L^p(R^d)) with 2/q +d/p < 1, that F ∈ L^q([0, T];W^2,p(R^d)). Hence, in the weak sense, F has 2 additional degrees of regularity compared to f in E. Thus, formula (1.1) tells us that the time-average of f with respect to the stochastic process (X_t^x)_t≥0 is more regular than f itself (it has 1 additional degree of regularity). This is what we call a stochastic regularization effect or regularization by noise. In this paper, we investigate the following open question stated in [17]:

"The generalization to nonlinear transport equations, wherebdepends onuitself, would be a major next step for applications to fluid dynamics but it turns out to be a difficult problem. Specifically there are already some difficulties in dealing with a vector field b which depends itself on the random perturbation W. There is no obvious extension of the Itô-Tanaka trick to integrals of the form RT

0 f(ω, s, X_s^x(ω))ds with randomf."

A major "pathology" in this problem is that there are simple examples of random functions f for which the Itô-Tanaka trick does not work anymore. As an example, consider a random functionf˜of the form

f˜(ω, s, x) :=f(x−W_s(ω)),

where (W_t)_t≥0 is the Brownian motion from (1.2). This gives, forb= 0 in (1.2), Z _T

0

f(ω, t, W˜ _t+x)dt= Z _T

0

f(t, x)dt, which does not bring any additional regularity.

It turns out that, whenf is a random function, the solutionF to (1.3) is not adapted anymore to F_t^W

t∈[0,T] the filtration of the Brownian motion, making the stochastic integral on the right-hand side of (1.1) ill-posed.

In this paper we tackle this difficulty by considering another equation which is the adapted version of the Fokker-Planck equation (1.3). More precisely, we show in Theo- rem 4.1 that given a random functionf which depends in an adapted way on a standard Brownian motion(W_t)_t≥0, the following formula holds

Z T 0

f(t, X_t^x)dt=−F(0, x)−

Z T 0

(∇F(s, X_s^x) +Z(s, X_s^x))dW_s− Z T

0

∇Z(s, X_s^x)ds, P−a.s.

(1.4) where (F, Z) is the predictable solution of the following backward stochastic partial differential equation (BSPDE)

F(t, x) =− Z T

t

1

2∆ +b(s, W_(s), x)· ∇

F(s, x)ds− Z T

t

f(s, x)ds− Z T

t

Z(s, x)dW_s, and (X_t^x)_t≥0 is a weak solution of the stochastic differential equation

X_t^x =x+ Z _t

0

b(s, W_(s), X_s^x)ds+W_t.

We name (1.4) theItô-Wentzell-Tanaka trick as the derivation of 1.4 call for the use of the Itô-Wentzell formula in place of the classical Itô formula which allows one to give a semimartingale type decomposition of F(t, X_t^x)when F(t, x) is itself a semimartingale random field. Note that we also allow b to depend on the Brownian motion W. This

contrasts with the classical Itô-Tanaka trick where bothf andbmust be deterministic mappings. The derivation of this formula calls for a study of the Fokker-Planck BSPDE.

In this direction, incidentally we prove new results as Theorem 3.1 on this equation in particular by allowing only L^q−L^p regularity on its coefficients together with a representation of its solution in terms of the solution to the non-adapted SPDE and of its Malliavin derivative by providing a methodology which generalizes: the well- knownlinearization technique used for linear BSDEs and deterministic semigroups (see [13, Proposition 2.2]), and a Feynman-Kac formula for BSPDEs related to Forward- Backward SDEs as in [25, Corollary 6.2]. We also prove that the solution processes (Y, Z) to the equation are Malliavin differentiable. The study of the BSPDE relies on the one of the non-adapted Fokker-Planck equation in Section 3.2.

There are well-known results concerning the regularization effects of stochastic process ondeterministicfunctions (see the survey of Flandoli [15]) but, to our knowledge, there exists no similar results in the case of random functions. The phenomenon is widely used in the recovery of the strong uniqueness of solutions of stochastic differential equations (SDE) with singular drifts [10, 20, 26, 24, 31, 34]. It has been generalized to SDE in infinite dimension [8, 9, 27] and the conditions for the existence of a stochastic flow has also drawn attention [1, 16, 32]. Another direction of interest is the improvement of the well-posedness of stochastic partial differential equations (SPDE). In particular, the stochastically perturbed linear transport equation has received a lot of interest [2, 4, 14, 17]. More recent works provide extensions to nonlinear SPDE, see for instance [3, 18, 19] for models from fluid mechanics and [6, 7, 11] for dispersive equations. Let us also mention that the type of processes that yield a regularization effect is not restricted to semi-martingales. For instance, in [30, 33] whereα-stable processes have been considered and, in [5], where the authors showed a regularization phenomenon using rough paths (in particular for the fractional Brownian motion).

The paper is organized as follows. In Section 2 we make precise the definitions and the notations that will be used later on. This includes some material on Malliavin calculus especially for random fields. Then, in Section 3 we introduce the transport SDE under interest and we study the adapted and the non-adapted Fokker-Planck equations. The Itô-Tanaka-Wentzell trick, together with an example, is presented in Section 4.

2 Notations and preliminaries

2.1 Main notations

Throughout this paper T will be a fixed positive real number and d denotes a fixed positive integer. For any x in R^d, we denote by |x| the Euclidian norm of x. Let (E,k · k_E)be a Banach space, we setB(E)the Borelianσ-field onE. For given Banach spacesE, F and anyp≥0, we setL^p(E;F)the set ofB(E)\B(F)-measurable mappings f :E →F such that

kfk^p_Lp(E;F):=

Z

kf(x)k^p_Fµ(dx)<+∞,

whereµis a non-negative measure on(E,B(E))(the Borelianσ-field onE). Naturally the norm depends on the choice of µ that will be made explicit in the context. If F = Rⁿ, n ∈ N, then we simply set L^p(E) := L^p(E;Rⁿ). We also denote by C⁰(E) (resp. C_b⁰(E)) the set of continuous (resp. bounded continuous) real-valued mappings f on E.

For any p >1we set p¯the Hölder conjugate of p.

For any mapping ϕ :R^d → R we denote by _∂x^∂ϕ

i the i-th partial derivative of ϕ (i = 1,· · · , n), by ∇ϕ:= (_∂x^∂ϕ

1, . . . ,_∂x^∂ϕ

d) the gradient of ϕ(when it is well-defined), and by

∆ϕits Laplacian. For a multi indexk:= (k1,· · · , k_d)inN^d, we set∇^kϕ:= ^∂k_∂x^{1+···+kdϕ}

1...∂^kd ϕ and |k|:=Pd

i=1k_i. Forp, m∈R⁺, we set

W^m,p(R^d) =n

ϕ∈L^p(R^d);F⁻¹

([1 +|ξ|²]^m/2ϕˆ

∈L^p(R^d))o , the usual Sobolev spaces equipped with its natural norm

kϕk_Wm,p(R^d):=

F⁻¹

([1 +|ξ|²]^m/2ϕˆ

L^p(R^d)),

where ϕ(ξ) =ˆ F(ϕ)(ξ) and F (resp. F⁻¹) denotes the Fourier transform (resp. the inverse Fourier transform). Let n, k ∈ N and α ∈ (0,1). We denote by C_b^k,α(E), the set of bounded functions having bounded derivatives up to order kand with α-Hölder continuouskth partial derivatives. It is equipped with the norm

kϕk_Ck,α

b (E):=kϕk_Ck

b(E)+ sup

|ℓ|=k

sup

x6=y

|∇^ℓf(x)− ∇^ℓf(y)|

|x−y|^α , where kϕk_Ck

b(E) := P

|ℓ|≤ksup_x∈E|∇^ℓf(x)|. Finally C₀^∞(Rⁿ), (n ∈ N^∗) stands for the set of infinitely continuously differentiable function with compact support.

Throughout this paper C will denote a non-negative constant which may differ from line to line.

Unless stated otherwise, we always assume that the real numbers p, q∈(2,∞) verify d

p+2 q <1.

2.2 Malliavin-Sobolev spaces

In this section we recall the classical definitions of Malliavin-Sobolev spaces presented in [28] and extended them to functional valued random variables that from now on we will refer as random fields. We start with some facts about Malliavin’s calculus for random variables.

2.2.1 Malliavin calculus for random variables

Let (Ω,F,P) be a probability space and W := (W_t)_t∈[0,T] a Brownian motion on this space (to the price of heavier notations all the definitions and properties in this section and of the next one extend to a d-dimensional Brownian motion). We assume that F =σ(W_t, t∈[0, T]).

LetS^rv be the set of cylindrical functionals, that is the set of random variable β of the form:

β =ϕ(W_t1,· · · , W_tn)

withN^∗,ϕ:Rⁿ →Rin C₀^∞(Rⁿ) and 0≤t₁ <· · ·< t_n≤T. For an element β in S^rv, we setDF the L²([0, T])-valued random variable as:

D_θβ :=

Xn i=1

∂ϕ

∂xi

(W_t1,· · · , W_tn)1_[0,ti](θ), θ∈[0, T].

For a positive integer p≥1, we setD^1,p the closure of S^rv with respect to the norm:

kβk^p_D1,p :=E[|β|^p] +E

"Z T 0

|D_θβ|²dθ p/2#

.

To Dis associated a dual operator denotedδ defined through the following integration by parts formula:

E[βδ(u)] =E Z T

0

Dtβ utdt

, (2.1)

for any β inD^1,2 and anyL²([0, T])-valued random variable u such that there exists a positive constantCsuch that

EhR_T

0 D_tγu_tdti

≤Ckγk_D1,2, ∀γ ∈D^1,2. In particular if u := (ut)_t∈[0,T] is a predictable process then δ(u) = RT

0 utdWt. In addition, according to [28, Proposition 1.3.4], for any β inS and anyh inL^p([0, T]) (withp≥2),δ(hβ) is well-defined and satisfies

δ(hβ) =βδ(h)− Z _T

0

h_tD_tβdt. (2.2)

2.2.2 Malliavin calculus for random fields

We now extend these definitions to random fields that is to measurable mappings F : Ω×R^d→R. More precisely, we consider S be the set of cylindrical fields, that is the set of random fieldsF of the form:

F =ϕ(W_t1,· · · , W_tn, x)

withϕ:Rⁿ×R^d→ R inC₀^∞(R^n+d). We fix p an integer with p≥2. For an element F inS, we set DF theL^p([0, T])-valued random field as:

D_θF :=

Xn i=1

∂ϕ

∂x_i(W_t1,· · · , W_tn, x)1_[0,ti](θ), θ∈[0, T].

Note that forF inS,D∇^kF =∇^kDF for any multi indexk. In addition, an integration by parts formula for the operators D∇^k can be derived as follows.

Lemma 2.1. Let F in S, h in L^p([0, T]) and G in S. Let k be a multi-index in N^d, then the following integration by parts formula holds true:

E Z T

0

Z

R^d

D_t∇^kF(x)h_tG(x)dxdt

=E Z

R^d

F(x)δ((∇^k)^∗G(x)h)dx

, (2.3)

where (∇^k)^∗ denotes the dual operator of∇^k.

Proof. By the Malliavin-integration by parts formula (seee.g. [28, Lemma 1.2.1]) and by the classical integration by parts formula in R^dwe have that:

E Z T

0

Z

R^d

D_t∇^kF(x)h_tG(x)dxdt

= Z

R^d

E Z T

0

D_t∇^kF(x)h_tG(x)dt

dx

= Z

R^d

Eh

∇^kF(x)δ(G(x)h)i

dx, by (2.1)

= Z

R^d

Eh

∇^kF(x)G(x)δ(h)i dx−

Z

R^d

E

∇^kF(x) Z T

0

D_tG(x)h_tdt

dx, by (2.2)

=E Z

R^d

F(x)(∇^k)^∗G(x)dxδ(h)

−E Z

R^d

F(x)(∇^k)^∗ Z T

0

D_tG(x)h_tdtdx

=E Z

R^d

F(x)

(∇^k)^∗G(x)δ(h)− Z T

0

Dt(∇^k)^∗G(x)htdt

dx

=E Z

R^d

F(x)δ((∇^k)^∗G(x)h)dx

, by (2.2).

This integration by parts formula allows us to prove that the operators D∇^k are closable.

Lemma 2.2. Letp≥2 andkbe in N^d. The operatorsD∇^k (and so∇^kD) are closable fromS to L^p(Ω×R^d;L^p([0, T])).

Proof. Let(Fn)⊂ Sa sequence of random fields which converges inL^p(Ω×R^d;L^p(R^d)) to 0 and such that (D∇^kF_n)_n converges in L^p(Ω×R^d;L^p([0, T])) to some element η in L^p(Ω×R^d;L^p([0, T])). Let h in L^p([0, T]) and G : R^d → R in S. We recall that

¯

p:= _p−1^p . For anyn≥1, it holds that

E Z

R^d

Z T 0

η(t, x)h_tdtG(x)dx

=E Z

R^d

Z T 0

(η(t, x)−D_t∇^kFⁿ(x))h_tdtG(x)dx

+E Z

R^d

Z T 0

D_t∇^kFⁿ(x)h_tdtG(x)dx

=E Z

R^d

Z T 0

(η(t, x)−Dt∇^kFⁿ(x))htdtG(x)dx

+E Z

R^d

Fⁿ(x)δ((∇^k)^∗G(x)h)dx

, where we have used the integration by parts formula (2.3). We estimate the two terms above separately. For the first one, using successive Hölder’s Inequality, we have that

E

Z

R^d

Z T 0

(η(t, x)−D_t∇^kFⁿ(x))h_tdtG(x)dx

≤E Z

R^d

Z T 0

|η(t, x)−Dt∇^kFⁿ(x)|^pdtdx 1/p

E[kGk^p_L^¯p¯(R^d)]^1/p¯khk_L^p¯_([0,T])

n→+∞−→ 0.

The second term can be estimated as follows (using also Hölder’s inequality and (2.2)).

E

Z

R^d

Fⁿ(x)δ((∇^k)^∗G(x)h)dx

= E

Z

R^d

Fⁿ(x)(∇^k)^∗G(x)δ(h)dx

−E Z

R^d

Fⁿ(x) Z T

0

D_t(∇^k)^∗G(x)h_tdtdx

≤CE Z

R^d

|Fⁿ(x)|^pdx 1/p

E

δ(h)^2¯p1/(2¯p)

∨ khk_L²_([0,T])

×



Eh

k(∇^k)^∗Gk^2¯_L^p_{2 ¯p(Rd)}i1/(2¯p)

+E



 Z

R^d

Z T 0

|D_t(∇^k)^∗G(x)|²dt ^p₂^¯

dx





1/¯p



n→+∞−→ 0.

Combining the previous estimates and relations we conclude that E

Z

R^d

Z T 0

η(t, x)h_tdtG(x)dx

= 0.

The conclusion follows from the fact that the set of elements of the formGh withh in L^p([0, T]) andG inS is dense in L^p(Ω×R^d;L^p([0, T])).

Remark 2.1. Lemma 2.1 and Lemma 2.2 still holds if we replace the differential op- erator ∇^k with the Bessel potential (1−∆)^m/2 for any m∈R⁺.

For a positive integer m, we setD^1,m,p the closure of S with respect to the norm:

kFk^p_D1,m,p :=kFk^p_Wm,p + Z T

0

Eh

kD_θFk^p_Wm,p(Rd)i

dθ, (2.4)

where we denote

kFk^p_Wm,p :=Eh

kFk^p_Wm,p(Rd)i . In addition, forF inD^1,m,p, we have sincep≥2:

kFk^p_D1,m,p ≥ Z

R^d

kFk^p_D1,pdx+ X

|k|≤m

Z

R^d

k∇^kFk^p_D1,pdx,

with equality if p= 2.

Remark 2.2. In particular, if a random field F belongs to D^1,m,p, then for a.e. (t, x), ω 7→ ∇^kF(t, x)(ω) belongs to the classical Malliavin space D^1,p whose definition has been recalled in Section 2.2.1 (for any k such that |k| ≤m) for random variables that depend only onω and not on(t, x).

We conclude this section on the Malliavin derivative by introducing the spaceD^1,m,p_q :=

L^q([0, T];D^1,m,p) (with p, q ≥2) which consists of mappings F : [0, T]×Ω×R^d → R such that

kFk^q

D^1,m,pq :=

Z T 0

kF(t,·)k^q_D1,m,pdt <+∞.

Furthermore, we extend the definition W^m,p-norm accordingly kFk^q_Wm,p

q =

Z T 0

kF(t,·)k^q_Wm,pdt.

3 Fokker-Planck SPDEs and BSPDEs

In this section, we study the transport SDE of interest together with two related Fokker- Planck equations. The first one which will be considered in Section 3.2 and will be referred as the non-adapted (or SPDE) Fokker-Planck equation. The second one will be called theadapted (or BSPDE) Fokker-Planck equation associated to the SDE, and will be introduced and studied in Section 3.3. This equation will be fundamental to derive in Section 4 the stochastic counterpart of the Itô-Tanaka trick (that we will name then Itô-Tanaka-Wentzell trick).

3.1 A SDE with random drift

In analogy to [17, 24, 26], we consider the following SDE:

Xt=X0+ Z t

0

b(s, W_(s), Xs)ds+Wt, t∈[0, T], (3.1) where bis assumed to be aB([0, T]× C([0, T])×R^d)-measurable map, X₀ is inR^dand W is ad-dimensional Brownian motion. To begin with, let us recall the definition of a weak solution to Equation (3.1).

Definition 3.1. A weak solution is a triple (X, W), (Ω,F,P), (F_t)_t∈[0,T] where

• (Ω,F,P)is a probability space equipped with some filtration(F_t)_t∈[0,T]that satisfies the usual conditions,

• X is a continuous, (F_t)_t∈[0,T]-adapted R^d-valued process, W is a d-dimensional (F_t)_t∈[0,T]-Wiener process on the probability space,

• P(X(0) =X₀) = 1 and P(R_t

0|b(s, W_(s), X_s)|ds <+∞) = 1, ∀t∈[0, T],

• Equation (3.1) holds for all t in [0, T] with probability one.

Assumption 3.1. There exists a weak solution (X, W), (Ω,F,P), (F_t)_t∈[0,T] to the SDE (3.1).

By defintion, W is a(F_t)_t∈[0,T]-Brownian motion. So we denote by (F_t^W)_t∈[0,T] its natural completed right-continuous filtration which satisfiesF_t^W ⊂ F_tfor anyt∈[0, T].

In the following, the spaces D^r,m,p or D^r,m,p_q are understood to be defined with respect to (Ω,F_T^W,P).

We now give a simple proof of existence and uniqueness of a weak solution to (3.1) under some non-optimal assumptions.

Proposition 3.1. Let b ∈ C_b¹(R^d;L^q([0, T];L^p(R^d)). Then there exists a unique weak solution to the SDE

X_t=X₀+ Z _t

0

b(s, W_s, X_s)ds+W_t, t∈[0, T]. (3.2) Proof. The proof is based on the Girsanov’s theorem. Let us first remark thatL(t, x) :=

sup_y∈Rd|∇_yb(t, y, x)|and˜b(t, x) := sup_y∈Rd|b(t, y, x)|belong inL^q([0, T];L^p(R^d)). Thus, since 2/q+d/p <1, by [24, Lemma 3.2] we have, ∀κ∈R⁺ and k= 1,2,

Eh

e^{κR0TL(s,Ws)kds}i +Eh

e^{κR0T˜b(s,Ws)kds}i

<+∞, (3.3)

where W is a standard Brownian motion.

Let (Xt)t≥0 a standard Brownian motion on a probability space (Ω,F,P) equipped with the filtration (F_t)_t∈[0,T]. We consider the following SDE

Y_t=Y₀− Z t

0

b(s, Y_t, X_t)ds+X_t, t∈[0, T]. (3.4) In this step, we prove that there exists a unique solution to (3.4). Sinceb is Lipschitz, the uniqueness is obtain by a Gronwall lemma. Moreover, by using classical a priori estimates for Lipschitz SDE, we obtain

E

"

sup

t∈[0,T]

|Y_t|²

#

≤C

|Y₀|²+T+E Z T

0

|b(s,0, X_s)|²+L(s, W_s)² ds

,

which yields the existence of a strong solution.

By (3.3), we have, ∀κ∈R⁺, Eh

e^{κR0T|b(s,Ys,Xs)|2ds}i

≤Eh

e^{κR0T˜b(s,Xs)2ds}i

<+∞.

We deduce that

ρ(·) :=e^{R0·b(s,Ys,Xs)ds−12R0·|b(s,Ys,Xs)|2ds},

is a martingale under P by Novikov’s criterion. Hence, by Girsanov’s theorem, the process Y is a Brownian motion under the measure Q given by ^dQ_dP =ρ(T). Thus, by rewritingY asW, the triple(X, W),(Ω,F,Q),(F_t)_t∈[0,T]is a weak solution to the SDE (3.2).

We will need below several technical results that we present now. In the following, we denote by (P_t,s)_s≥t≥0 the heat semigroup.

Lemma 3.1. Let 1 < q, p < +∞. Then, there exists a constant C such that, ∀f ∈ D^1,0,p_q ,

Z _T

t

P_t,sf(s, x)ds

D^1,2,pq

≤Ckfk_D^1,0,p

q , (3.5)

and, another constant C_T >0 such that, ∀ε >0 and∀φ∈D1,2−2/q+ε,p, kPt,Tφk_D^1,2,p

q ≤CTkφk_D1,2−2/q+ε,p. (3.6)

Proof. Second estimate: The second estimate is a direct consequence of the a similar estimate in deterministic spaces. It is based on the following lemma [21, Theorem 7.2].

Lemma 3.2. Let v ∈ L^p(R^d), α ∈ [0,1], and t > 0. There exists a constant C > 0 such that

ke^−tP0,tvk_W2α,p(R^d)≤ C

t^αkvk_Lp(R^d) and k(P0,t−1)vk_Lp(R^d)≤Ct^2αkvk_W2α,p(R^d). (3.7) By setting α = 1/q−ε/2 and v = (1−∆)^m/2φ, with m= 2−2/q+ε, in Lemma 3.2, we obtain

kP_t,Tφk_D1,2,p ≤ Ce^T−t

(T −t)^1/q−ε/2kφk_D1,2−2/q+ε,p, thus, this yields the desired estimate

kPt,Tφk_D^1,2,p

q ≤C

Z T 0

e^qτ τ^1−qε/2dτ

1/q

kφk_D1,2−2/q+ε,p. (3.8)

First estimate: For the first estimate, the arguments of the proof are similar to those of [22, Theorem 1.1]. First, let us remark that in the case p=q, Estimate (3.5) can be deduce directly by using the classical inequality

Z _T

t

P_t,sg(s, x)ds

L^p([0,T],W^2,p)

≤Ckgk_Lp([0,T]×R^d), ∀g∈L^p([0, T]×R^d).

Therefore, it remains to prove estimate (3.5) for q 6= p. To do this, we apply the Calderón-Zygmund Theorem in Banach spaces (see [22, Theorem 1.4] for a precise statement). More precisely, we define the operator

Af(t, x) :=

Z

R

P_t,sf(s, x)1t≤s≤Tds,

which is a bounded operator from L^p(R,D^1,0,p) to L^p(R,D^1,2,p) since Estimate (3.5) is valid forq =p. Therefore, to apply the Calderón-Zygmund Theorem, we only need to prove the following estimate,∀t6=s,

k∂_s^ℓP_t,sfk_D1,2,p ≤ C

(s−t)²kfk_D1,2,p, (3.9)

for ℓ= 0,1, which can be deduced from the classical inequality k∇^kP_t,sfk_Lp(R^d)≤ C

(s−t)^|k|/2kfk_Lp(R^d), (3.10) and the fact that ∂_sP_t,s = ¹₂∆P_t,s. This enables us to extend the operator A to a bounded operator from L^q(R,D^1,0,p) to L^q(R,D^1,2,p), ∀q ∈ (1, p]. Finally, we remark that the adjoint operator ofA is given by

A^∗f(t, x) = Z t

0

P_s,tf(s, x)ds.

Thus, we are able to apply the same results toA^∗and conclude thatAis also a bounded operator fromL^q¯(R,D^1,0,¯p) toL^¯q(R,D^1,2,¯p),∀q∈(1, p]. This extends the range of q to (1,∞) for A.

The next result gives a Schauder estimate on the solution of a backward heat equa- tion with a source term inD^1,0,p_q . Its proof is similar to the one from [21, Theorem 7.2]

and the arguments can be directly extended to the norms D^1,m,p_q .

Proposition 3.2. Let 1 < q, p < +∞, 2/q < β ≤ 2 and f ∈ D^1,0,p_q . Denote, for (t, x)∈[0, T]×R^d,

u(t, x) :=− Z T

t

P_t,sf(s, x)ds.

Then, there exists a constantC >0 independent ofT such that, for any0≤s≤t≤T, ku(t)−u(s)k_D1,2−β,p ≤C(t−s)^β/2−1/qkfk_D^1,0,p

q , (3.11)

and, thus,

kukC_b^0,β/2−1/q([0,T];D^1,2−β,p)≤Ckfk_D1,0,p

q . (3.12)

A direct consequence of the previous result is the following Corollary 3.1. Let f ∈D^1,0,p_q . Denote, for (t, x)∈[0, T]×R^d,

u(t, x) :=− Z T

t

P_t,sf(s, x)ds.

Then, for anyε∈(0,1) satisfying

ε+d p +2

q <1,

there exists a constant C >0 and ε >˜ 0 such that, ∀t∈[0, T],

E

ku(t,·)k^p

C_b^1,ε(R^d)

+E

Z T 0

kD_θu(t,·)k^p

C_b^1,ε(R^d)dθ 1/p

≤C(T−t)^ε/2˜ kfk_D1,0,p

q .

(3.13) Proof. Let β = ˜ε+ 2/q where 0<ε <˜ 1−(ε+d/p+ 2/q). The result follows by the Sobolev embeddingC_b^1,α⊂W^2−β,p, withα= 1−β−d/p= 1−ε˜−q/2−d/p > ε, and Proposition 3.2.

3.2 The non-adapted Fokker-Planck equation

We set the linear operator L^X_t onC₀^∞(R^d):

L^X_t ϕ(x) := 1

2∆ϕ(x) +b(t, x)· ∇ϕ(x), and consider here the non-adapted Fokker-Planck equation

F(t, x) =φ(x)− Z _T

t

L^X_r F(r, x)dr− Z _T

t

f(r, x)dr. (3.14) Definition 3.2. A strong solution to Equation (3.14) is a function F in D^1,2,p_q such that, for allt∈[0, T], we have

F(t, x) =φ(x)− Z T

t

L^X_r F(r, x)dr− Z T

t

f(r, x)dr. (3.15) Remark 3.1. Note that each random variable F(t,·) solution to the previous SPDE is FT-measurable, and hence it is not adapted (compare with Remark 3.2 below).

We provide a Malliavin differentiability analysis for the solution the Fokker-Planck equation (3.14). We define,∀m≥0,

G^1,m,p_q :=

F ∈D^1,m,p_q ;∂tF ∈D^1,0,p_q , and the associated norm

kFk_G1,m,p

q :=kFk_D1,m,p

q +k∂_tFk_D1,0,p

q .

We begin with a result concerning the existence and uniqueness of a solution to the non-adapted Fokker-Planck equations.

Assumption 3.2. We assume that there exists a function ˜b ∈ L^q([0, T]) such that,

∀(t, w)∈[0, T]× C([0, T]), kb(t, w,·)k_Lp(R^d)+

Z T 0

kD_θb(t, w,·)k^p_Lp(Rd)dθ 1/p

≤˜b(t).

Lemma 3.3. Assume that 3.2 is in force. Let u∈G^1,2,p_q and denote ku(t,·)k^p_H1,p :=E

"

sup

x∈R^d

|∇u(t, x)|^p

# +E

"Z T 0

sup

x∈R^d

|∇D_θu(t, x)|^pdθ

# .

The followings estimates hold sup

t∈[0,T]

ku(t,·)k_H1,p ≤C_Tkuk_G1,2,p

q , (3.16)

whereCT is uniformly bounded with respect toT in compact sets ofR⁺, and,∀t∈[0, T], kb(t,·)· ∇u(t,·)k_D1,0,p ≤C˜b(t)ku(t,·)k_H1,p. (3.17) Proof. Firstly, let us remark that we have, ∀u∈G^1,2,p_q ,

u(t, x) =− Z T

t

P_t,r

∂_tu(r, x)−1

2∆u(r, x)

dr, and then, by using Corollary 3.1, we obtain the estimate

sup

t∈[0,T]

ku(t,·)k_H1,p ≤C_Tkuk_G1,2,p

q .

Secondly, we compute kb(t,·)· ∇u(t,·)k^p_D1,0,p =Eh

kb(t,·)· ∇u(t,·)k^p_Lp(R^d)

i +

Z _T

0

Eh

kD_θb(t,·)· ∇u(t,·) +b(t,·)·D_θ∇u(t,·)k^p_Lp(Rd)i dθ

≤Eh

kb(t,·)· ∇u(t,·)k^p_Lp(R^d)

i +CE

Z T 0

kD_θb(t,·)· ∇u(t,·)k^p_Lp(Rd)dθ

+CE Z T

0

kb(t,·)·D_θ∇u(t,·)k^p_Lp(Rd)dθ

.

Since the Malliavin derivative commutes with the spatial derivative inL^p, we obtain kb(t,·)· ∇u(t,·)k^p_D1,0,p ≤E

"

kb(t,·)k^p_Lp(Rd) sup

x∈R^d

|∇u(t, x)|^p

#

+CE

"Z T 0

kD_θb(t,·)k^p_Lp(R^d)dθ sup

x∈R^d

|∇u(t, x)|^p

#

+CE

"

kb(t,·)k^p_Lp(R^d)

Z _T

0

sup

x∈R^d

|∇D_θu(t, x)|^pdθ

# . Thus, by Assumption 3.2, we have (3.17) as

kb(t,·)· ∇u(t,·)k_D1,0,p ≤ C˜b(t) E

"

sup

x∈R^d

|∇u(t, x)|^p

# +E

"Z T 0

sup

x∈R^d

|∇D_θu(t, x)|^pdθ

#!1/p

.

Proposition 3.3. Let f ∈D^1,0,p_q andφ∈D1,2−2/q+ε,p, with ε >0. Under Assumption 3.2, there exists a unique solution F in G^1,2,p_q to the equation

F(t, x) =P_t,Tφ(x)− Z _T

t

P_t,sf(s, x)ds− Z _T

t

P_t,s[b(s, x)· ∇F(s, x)]ds. (3.18)

Moreover, the following estimate on the solution holds

kFk_G^1,2,p

q ≤CT

kφk_D1,2−2/q+ε,p +kfk_D^1,0,p

q

, (3.19)

where C_T > 0 depends on k˜bk_L^q_([0,T]) and is uniformly bounded with respect to T on compact sets ofR⁺.

Proof. Step 1: By using Corollary 3.1 and (3.17), we have kF(t,·)k^q_H1,p ≤CkP_t,Tφk^q_H1,p+C_Tkfk^q

D^1,0,pq +C_Tkb· ∇Fk^q

D^1,0,pq

≤Ckφk^q_H1,p+C_Tkfk^q

D^1,0,pq +C_T Z T

t

|˜b(s)|^qkF(s,·)k^q_H1,pds.

Thanks to a Gronwall lemma and the Sobolev embeddingC_b^1,ε ⊂W^2−2/q+ε,p, we deduce sup

t∈[0,T]

kF(t,·)k_H1,p ≤

C_Tkfk_D1,0,p

q +Ckφk_D1,2−2/q+ε,p

e^{CT Tq k˜bkqLq([0,T])}. (3.20) We now turn to Estimate (3.19). We can apply the D^1,2,p_q -norm to (3.18) and obtain, by using lemma 3.1,

kFk^q

D^1,2,pq ≤C_Tkφk^q_D1,2−2/q+ε,p +Ckfk^q

D^1,0,pq +C Z T

t

kb(s,·)· ∇F(s,·)k^q_D1,0,pds

≤CTkφk^q_D1,2−2/q+ε,p +Ckfk^q

D^1,0,p_q +C Z T

t

|˜b(s)|^qkF(s,·)k^q_H1,pds which yields, thanks to (3.20),

kFk^q

D^1,2,p_q ≤CT(1 +k˜bk^q_Lq([0,T])e^{CTTk˜bkqLq([0,T])})

kφk^q_D1,2−2/q+ε,p +kfk^q

D^1,0,p_q

, (3.21) Then, we differentiate (3.18) with respect to the time variable and deduce the equation

∂tF(t, x) =L^X_t F(t, x) +f(t, x),

F(T, x) =φ(x). (3.22)

By applying the D^1,0,p_q -norm to (3.22) and by using the estimate (3.20), we obtain k∂_tFk_D1,0,p

q ≤ 1

2k∆Fk_D1,0,p

q +kfk_D1,0,p

q +kb· ∇Fk_D1,0,p q

≤C_T

kφk_D1,2−2/q+ε,p +kfk_D1,0,p q

, which, together with (3.21), gives Estimate (3.19).

Step 2: The last argument of the proof consists in using the so-called continuity method. For µ∈[0,1], we consider the equation

F_µ(t, x) =P_t,Tφ(x)− Z T

t

P_t,sf(s, x)ds− Z T

t

P_t,s[µb(s, x)· ∇F_µ(s, x)]ds. (3.23) We wish to prove that the setν ⊂[0,1]of elementsµfor which (3.23) admits a unique solution is [0,1] (with µ= 1corresponding to the equation (3.18)). In the case where µ = 0, the existence and uniqueness of a solution of (3.18) is straightforward and, thus, ν is not empty. Fix µ₀ ∈ ν and denote R^µ0 the mapping from D^1,0,p_q to G^1,2,p_q

which maps f to the solution F_µ0 of (3.23) for φ = 0. Let µ ∈ [0,1] to be fix later.

The existence and uniqueness of the solution of equation (3.23) relies on a fixed point argument. We consider the mappingΓ_µ given by

Γ_µ(F) =P_·,Tφ+R^µ0f+ (µ−µ₀)R^µ0(b· ∇F),

and aim to prove that it is a contraction mapping fromG^1,2,p_q to itself. It follows from the estimates (3.19) and (3.16) that, ∀F₁, F₂ ∈G^1,2,p_q ,

kΓ_µ(F₁)−Γ_µ(F₂)k_G^1,2,p

q ≤C|µ−µ₀|kb· ∇(F₁−F₂)k_D^1,0,p

q

≤C|µ−µ₀| Z T

0

|˜b(s)|^qkF₁(s,·)−F₂(s,·)k^q_C1,pds 1/q

≤C|µ−µ₀|k˜bk_Lq([0,T])kF₁−F₂k_G1,2,p

q .

Hence, by choosingµsuch that|µ−µ₀|< ¹

Ck˜bk_Lq([0,T]), we can conclude that there exists a unique solution to (3.23). Therefore, by repeating the argument a finite number of times, we prove that ν = [0,1]and that (3.18) admits a unique solution inG^1,2,p_q . Corollary 3.2. Let f ∈ D^1,0,p_q and φ∈ D1,2−2/q+ε,p, with ε > 0. Under Assumption 3.2, there exists a unique solution F in D^1,2,p_q to the equation (3.14).

Proof. The existence of the solution follows directly from Proposition 3.3 since one can check that a solution of (3.18) is a solution to (3.14). To prove the uniqueness, we consider a solutionF of (3.14) withφ= 0and f = 0. Let Fⁿbe a sequence of smooth functions in(t, x) ofG^1,2,p_q such that

kF −Fⁿk_D1,2,p

q +k∂_tF −∂_tFⁿk_D1,0,p

q −→

n→∞0.

Therefore, we have that

∂_tFⁿ(t, x)− L^X_t Fⁿ(t, x) −→

n→∞∂_tF(t, x) +L^X_t F(t, x) = 0,

in D^1,0,p_q . By denoting R the linear bounded operator from D^1,0,p_q to G^1,2,p_q which associates f with the solutionF of (3.18) and sinceRf solves (3.14), we have a repre- sentation of Fⁿ as

Fⁿ=R ∂_tFⁿ− L^XFⁿ

. (3.24)

It follows from (3.24) and (3.19) that kFⁿk_G1,2,p

q ≤Ck∂_tFⁿ− L^XFⁿk_D1,0,p

q −→

n→∞0, which implies that kFk_G^1,2,p

q = 0.

From now on, we denote (P_s,t^X)_{0≤s≤t≤T} the family of linear operators associated to the solution of the Fokker-Planck equation determined by L^X, that is,P_s,t^Xφ(x) is the solution to the SPDE

P_s,t^Xφ(x) =φ(x)− Z t

s

L^X_r P_r,t^Xφ(x)dr, 0≤s≤t, (3.25) withφa F_t-measurable mapping inD1,2−2/q+ε,p. We end this section by the following Lemma which gives some estimates onP^X.

Lemma 3.4. Let f ∈D^1,0,p_q and φ∈D1,2−2/q+ε,p, with ε >0. Under Assumption 3.2, the following estimates hold

kP_·,T^Xφk_G1,2,p

q ≤C_1,Tkφk_D1,2−2/q+ε,p, (3.26)

Z T

·

P_·,r^Xf(r,·)dr

G^1,2,pq

≤C2,Tkfk_D^1,0,p

q , (3.27)

and Z T

0

kL^X_· P_·,r^Xf(r,·)k^q

D^1,0,p_q dr≤Ckfk^q

D1,2−2/q+ε,p q

. (3.28)

Proof. The estimates (3.26) and (3.27) are direct consequences of Proposition 3.3. For the last estimate, thanks to (3.16), (3.17), and (3.26), there exists a constant C_r >0 uniformly bounded inr ∈[0, T]such that

kb· ∇P_·,r^Xf(r,·)k_D1,0,p

q ≤C_rkf(r,·)k_D1,2−2/q+ε,p. Therefore, the estimate follows from (3.26) since

Z T 0

kL^X_· P_·,r^Xf(r,·)k^q

D^1,0,p_q dr≤ Z T

0

C_r^qkf(r,·)k^q_D1,2−2/q+ε,pdr.

We can also compute the Malliavin derivative of (P_s,t^X)_{0≤s≤t≤T}. This is goal of the next lemma.

Lemma 3.5.We have the following commutation formula between the Malliavin deriva- tive and the operator P^X

D_tP_t,T^X φ(x) =P_t,T^XD_tφ(x)− Z T

t

P_t,r^X D_tb(r, x)· ∇P_r,T^X φ(x)

dr. (3.29) Proof. Let t≤r ≤T. Denote

Φ(r, x) :=DtP_r,T^X φ(x),

then, a direct computation of the Malliavin derivative applied to the representation formula ofP^X gives

Φ(r, x) = Φ(T, x)− Z T

r

L^X_uΦ(u, x)du− Z T

r

D_tb(u, x)· ∇P_u,T^X φ(x)du.

Hence, by the representation formula ofP^X, we deduce the followingmild formulation ofΦ

Φ(r, x) =P_r,T^X Φ(T, x)− Z _T

r

P_r,u^X D_tb(u, x)· ∇P_u,T^X φ(x) du, and, thus, the desired result.

3.3 The adapted Fokker-Planck equation

We consider now the following BSPDE:

F(t, x) =− Z _T

t

L^X_r F(r, x) +f(r, x) dr−

Z _T

t

Z(r, x)dWr, (3.30) where f belongs to D^1,0,p_q . To ensure the existence of such representation, we need to work under the natural filtration (F_t^W)_t∈[0,T] of the Brownian motion W. In this section we will call a predictable process a (F_t^W)_t∈[0,T]-predictable stochastic process.

Note that by the definition of a weak solution (cf. Definition 3.1) as F_·^W ⊂ F_·, any (F_t^W)_t∈[0,T]-predictable process is (F_t)_t∈[0,T]-predictable.

From now on we assume that:

Assumption 3.3. f is a predictable field.

Before going further we recall what is a solution to the BSPDE (3.30) in our context.

To this end we say that a random fieldϕ: Ω×[0, T]×R^d→Ris predictable if for any x inR^d,ϕ(·, x) is predictable. We set form∈N:

W^m,p_P,q :={ϕpredictable field , kϕk_W^m,p_q <+∞}, M^p:=

(

ϕpredictable field , Z

R^d

E

"Z T 0

|ϕ(s, x)|²dt ^p₂#

dx <+∞

) .

Definition 3.3 (Adapted strong solution to a BSPDE). We say that a pair of pre- dictable random fields (F, Z) is strong solution to the BSPDE (3.30)if

(F, Z)∈W^2,p_P,q×M^p

with ^d_p +²_q <1 and Relation (3.30) is satisfied for every t in [0, T], for a.e. x in R^d, P-a.s..

Remark 3.2. We warn the reader that in the previous definition, the predictable feature of the fields(F, Z) is crucial. In that sense we will speak of BSPDE. This differs from the SPDE (3.14) whose solution is not adapted (see Remark 3.1). In that case we will speak of SPDEs to emphasis that the measurability requirement is not present.

In order to proceed further, we need some additional assumptions on the Malliavin derivatives off andb.

Assumption 3.4. Let m∈[q,∞] andℓ∈[p,∞]such that 1

m+ 1

¯ m = 1

q and 1

ℓ + 1 ℓ¯= 1

p.

We assume that there exist a function f^′ ∈ L^m([0, T];L^ℓ(Ω;L^p(R^d))) (resp. b^′) and a function m_f ∈L^m¯([0, T];L^ℓ¯([0, T]×Ω)) (resp. m_b) such that

D_θf(t, x) =f^′(t, x)m_f(θ, t)

Moreover, we assume that, for a.e. t∈[0, T], ∂_tm_f(t,·) (resp. ∂_tm_b(t,·)) is a measure on [0, T]and that there exists a constant C >0 such that

Z T

·

P_·,s^Xf^′(s,·)∂_tm_f(·, ds) _W0,p

q

≤Ckf^′k_W0,p q .

Finally, we assume that Tr(m_f)(t) :=m_f(t, t)(resp. Tr(m_b)) belongs toL^m¯([0, T];L^¯ℓ(Ω)).