Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations

Introduction

This paper situates in the framework of forward probabilistic representations of nonlinear PDEs of the form

Coming back to (1.1), allowing $\Lambda\neq 0$ encompasses the case of Burgers-Huxley or Burgers-Fisher equations which are of great importance to represent nonlinear phenomena in various fields such as biology , physiology and physics . These equations have the particular interest to describe the interaction between the reaction mechanisms, convection effect, and diffusion transport. However our aim is also to consider (via time reversal) PDEs coming from stochastic control as non-linear HJB equations.

For (1.1), we propose the forward probabilistic representation

where $W$ is a $p$-dimensional Brownian motion. (1.2) is a nonlinear stochastic differential equation (NLSDE) in the spirit of McKean, see e.g. . The justification of the proposed probabilistic representation relies on the fact whenever a solution $(Y,u)$ of (1.2) exists then $u$ is a weak (in the sense of distributions) of (1.1); this follows in elementary way through an application of Itô formula.

The underlying idea of our approach consists in extending, to fairly general non-conservative PDEs, the probabilistic representation of nonlinear Fokker-Planck equations which appears when $\Lambda=0$. An interesting aspect of this strategy is that it is potentially able to represent an extended class of second order nonlinear PDEs.

When $\Lambda=0$, several authors have studied NLSDEs of the form (1.2). Significant contributions are due to , , , in the case where the non linearity with respect to $u$ are mollified in the diffusion and drift coefficients. In , the authors focused on the case when the coefficients depend pointwisely on $u$. The authors have proved strong existence and pathwise uniqueness of (1.2), when $\Phi$ and $g$ are smooth and Lipschitz and $\Phi$ is non-degenerate. Other authors have more particularly studied an NLSDE of the form

In this paper we will concentrate on the novelty constituted by the introduction of $\Lambda$ depending on $u$ and $\nabla u$. For this step $\Phi$, $g$ will not depend on $u$. In this context we will focus on semilinear PDEs of the form

with $L^{\ast}$ a partial differential operator of the type

If $\Lambda=0$ (1.4) is the classical Fokker-Planck equation. An alternative approach for representing this type of PDE is given by forward-backward stochastic differential equations. Those were initially developed in , see also for a survey and for a recent monograph on the subject. However, the extension of those equations to fully nonlinear PDEs still requires complex developments and is the subject of active research, see for instance . Branching diffusion processes provide another probabilistic representation of semilinear PDEs, see e.g. . Here again, extensions to second order nonlinear PDEs still constitutes a difficult issue.

As suggested, our method potentially allows to reach a certain significant class of PDEs with second-order non-linearity, if we allow the diffusion coefficient to also depend on $u$. The general framework where $g$ and $\Phi$ also depend non linearly on $u$ while $\Lambda$ depends on $u$ and $\nabla u$ has been partially investigated in , where the dependence of the coefficients with respect to $u$ is mollified and $\Lambda$ does not depend on $\nabla u$. An associated interacting particle system converging to the solution of a regularized version of the nonlinear PDE has been proposed in , providing encouraging numerical performances. The originality of the present paper is to consider a pointwise dependence of $\Lambda$ on both $u$ and $\nabla u$. The pointwise dependence on $\nabla u$ constitutes the major technical difficulty. For this we introduce a new approach based on the technique of mild solutions making use of the semigroupe associated with $L_{t}$. For this reason in this paper we concentrate on non-linearities only in $\Lambda$ leaving extensions in the forthcoming paper where we authorize the coefficient $b$ to depend on $u$

The theoretical analysis of the performance of the time-discretized algorithm related to the present paper has been performed in Theorem 3.4 in . In that paper we test the algorithm with respect to the Burgers and KPZ equations for which there are explicit solutions.

Preliminaries

$V\,:[0,T]\times{\mathcal{C}}^{d}\times{\cal C}\times{\cal C}^{d}$ is defined for any functions $x\in{\cal C}^{d}$, $y\in{\cal C}$ and $z\in{\cal C}^{d}$, by

Therefore, if $\Lambda$ is supposed to be bounded and Lipschitz w.r.t. to its space variables $(x,y,z)$, uniformly w.r.t. $t$, we observe that (2.2) implies that, for all $t\in[0,T]$, $x,x^{\prime}\in{\cal C}^{d}$, $y,y^{\prime}\in{\cal C}$, $z,z^{\prime}\in{\cal C}^{d}$,

$M_{\Lambda}$ (resp. $L_{\Lambda}$) denoting an upper bound of $|\Lambda|$ (resp. the Lipschitz constant of $\Lambda$), see also Assumption 2.

In the sequel, $K$ may be asked to additionally verify the following conditions.

2 Mild and Weak solutions

We first introduce the following assumption.

The functions $s\in[0,T]\mapsto|\Phi(s,0)|$ and $s\in[0,T]\mapsto|g(s,0)|$ are bounded.

Given any $\sigma(W_{r},r\leq s)$-measurable r.v. $Y_{s}$, classical theorems for SDE with Lipschitz coefficients imply strong existence and pathwise uniqueness for the SDE

Let $s=0,Y_{0}\sim{\bf u_{0}}$. From now on $Y$ will be the unique strong solution of the SDE

Its ”adjoint” $L^{\ast}_{t}$ defined in (1.5), verifies

is a solution in the sense of distributions to the Fokker-Planck equation

i.e., for all $\varphi\in C_{0}^{\infty},$

In particular (2.15) with $\nu_{0}=\delta_{x_{0}}$ says that

As mentioned in the introduction, a natural approach to show the link between (1.4) and (1.6) consists in applying Itô’s formula to the solution $Y$ of (2.10): if $(Y,u)$ is a solution of (1.6), then $u$ is a weak solution of (1.4). However, in this paper, instead of the notion of weak solution, we will make use of the notion of mild solution. The link between those two notions is discussed in the proposition below.

We assume that $\nu=0$ is the unique solution in the sense of distributions of (2.15) with $\nu_{0}=0$. Then, $u$ is a mild solution of (1.4) if and only if $u$ is a weak solution of (1.4)

Postponed to the Appendix, see Section 6.3.

There exist several sets of technical assumptions (see e.g. ) leading to the uniqueness assumed in Proposition 2.2 above. In particular, under items 1., 2. and 3. of Assumption 2 stated in Section 3 (which will constitutes our framework in the sequel), Theorem 4.7 in Chapter 4 of ensures (classical) existence and uniqueness of the solution of (2.15), see also Lemma 6.4 in the Appendix.

Feynman-Kac type representation

The first proposition below shows how the map $t\mapsto\mu(t,\cdot)$ can be characterized as a solution of the linear parabolic PDE

Before stating the corresponding proposition, we introduce the notion of measure-mild solution.

Under Assumption 1 the measure-valued map $\mu$ defined by (3.1) is the unique measure-mild solution of

where the operator $L_{t}^{\ast}$ is defined by (1.5).

We first prove that a function $\mu$ defined by (3.1) is a measure-mild solution of (3.4). Observe that for all $t\in[0,T]$,

Taking the supremum over $\varphi$ such that $\|\varphi\|_{\infty}\leq 1$ in each side of (3.11), we get

Gronwall’s lemma implies that $\nu(t,\cdot)=0$. Uniqueness of measure-mild solution for (3.4) follows. This ends the proof. ∎

The next lemma shows how a measure-mild solution of (3.4), which is a function defined on $[0,T]$ can be built by defining it recursively on each sub-interval of the form $[r,r+\tau]$. In particular, it will be used in Theorem 3.6 and Proposition 4.4. Its proof is postponed in Appendix (see Section 6.4).

for all $t\in[k\tau,(k+1)\tau]$ and $k\in\{0,\cdots,N-1\}$, if and only if $\mu$ is a measure-mild solution (in the sense of Definition 3.1) of (3.4).

A function $u$ verifying (3.15) will be called a Feynman-Kac type representation of (1.4).

We now precise more restrictive assumptions to ensure regularity properties of the transition probability function $P(s,x_{0},t,dx)$ used in the sequel.

$\Phi$ and $g$ belong to $C^{0,3}_{b}$. In particular, $\Phi$, $g$ are uniformly bounded and $M_{\Phi}$ (resp. $M_{g}$) denote the upper bound of $|\Phi|$ (resp. $|g|$).

$\Lambda$ is supposed to be uniformly bounded: let $M_{\Lambda}$ be an upper bound for $|\Lambda|$.

The lemma below establishes, under a suitable choice of $\tau>0$, existence and uniqueness of the mild solution on $[r,r+\tau]$, with initial condition $\phi$ at time $r$, i.e. existence and uniqueness of the fixed-point for the application $\Pi$.

We first insist on the fact that all along the proof, the dependence of $\widehat{u}_{0}$ w.r.t. $r,\phi$ in (3.20) will be omitted to simplify notations. Let us fix $r\in[0,T-\tau]$.

We first check that \Pi\Big{(}L^{1}([r,r+\tau],B(0,M))\cap B_{\infty}(0,M)\Big{)}\subset L^{1}([r,r+\tau],B(0,M))\cap B_{\infty}(0,M). Let us fix $v\in L^{1}([r,r+\tau],B(0,M))\cap B_{\infty}(0,M)$. For $t\in[r,r+\tau]$,

where we have used the fact that $x\mapsto p(s,x_{0},t,x)$ is a probability density, the boundedness of $\Lambda$ and the bounds $\|v(s,\cdot)\|_{1}\leq M$ and $\|\widehat{u_{0}}(s,\cdot)\|_{1}\leq M$ for $s\in[r,r+\tau]$.

where the constant $C_{u}$ and the Gaussian kernel $q$ come from inequality (6.15) and only depending on $\Phi$ and $g$. Consequently, taking into account (3.23) and (3.25), we obtain,

We deduce that $\Pi(v)\in L^{1}([r,r+\tau],B(0,M))\cap B_{\infty}(0,M)$. Let us fix $t\in[r,r+\tau]$, $v_{1},v_{2}\in L^{1}([r,r+\tau],B(0,M))\cap B_{\infty}(0,M)$. $\Lambda$ being bounded and Lipschitz, the notation introduced in (3.21) and inequality (2.3) imply

Iterating the procedure once again yields

for all $t\in[r,r+\tau]$. Interchanging the order in the second integral in the r.h.s. of (3), we obtain

where the latter line above comes from the fact for all $\theta>0$,

$\Gamma,B$ denoting respectively the Euler gamma and Beta functions. Injecting inequality (3.34) in (3), we obtain for all $t\in[r,r+\tau]$

Iterating previous inequality, one obtains the following. For all $k\geq 1$, $t\in[r,r+\tau]$,

By induction on $k\geq 1$ (3.36) can indeed be established. Finally, by integrating each sides of (3.36) w.r.t. $dt$ and using Fubini’s theorem, for $k\geq 1$, we obtain

where $\widehat{u_{0}}^{0}(t,x)$ is given by (3.19) with $\phi=u_{0}$, i.e.

Setting $u^{0}:=\widehat{u_{0}}^{0}+v^{0}$, i.e.

where the second inequality comes from (3.19) with $r=k\tau$ and $\phi=u^{k-1}(k\tau,\cdot)$.

In the case where the function $\Lambda$ does not depend on $\nabla u$, existence and uniqueness of a solution of (1.4) in the mild sense can be proved under weaker assumptions. This is the object of the following result.

Since this theorem can be proved in a very similar way as Theorem 3.6 but with simpler computations, we omit the details. ∎

Existence/uniqueness of the Regularized Feynman-Kac equation

The concept of mild solution associated to this type of equation is clarified by the following definition.

Similarly as Theorem 3.5, we straightforwardly derive the following equivalence result.

similarly to (3.19). In the sequel, the dependence of $\widehat{u}_{0}$ w.r.t. $r,\pi$ will be omitted when it is self-explanatory.

Assume the validity of items 4. and 5. of Assumption 2. Let $\pi\in B(0,M)$. Let us fix $\varepsilon>0$ and $M>0$ such that $M\geq\|\pi\|_{TV}$. Then, there is $\tau>0$ only depending on $M_{\Lambda}$ such that for any $r\in[0,T-\tau]$, $\Pi_{\varepsilon}$ admits a unique fixed-point in ${\cal B}([r,r+\tau],B(0,M))$.

Recalling (4), where $\widehat{u_{0}}$ is defined by (4.5), it follows that for all $\beta\in{\cal B}([r,r+\tau],B(0,M))$, $t\in[r,r+\tau]$,

where for the latter inequality of (4.7) we have used the definition of $\tau:=\frac{1}{2M_{\Lambda}}$. We deduce that $\Pi({\cal B}([r,r+\tau],B(0,M)))\subset{\cal B}([r,r+\tau],B(0,M))$. Consider now $\beta^{1},\beta^{2}\in{\cal B}([r,r+\tau],B(0,M))$. For all $\lambda>0$ we have

with $C_{\varepsilon,T}:=2L_{\Lambda}M(\|K_{\varepsilon}\|_{\infty}+\|\nabla K_{\varepsilon}\|_{\infty})+M_{\Lambda}$. It follows

Hence, taking $\lambda>C_{\varepsilon,T}$, $\Pi_{\varepsilon}$ is a contraction on ${\cal B}([r,r+\tau],B(0,M))$.

and for all $k\in\{1,\cdots,N\}$, $t\in[k\tau,(k+1)\tau]$,

Previous discussion leads us to the following proposition.

Suppose the validity of Assumption 1 and items 4. and 5. of Assumption 2. Let us fix $\varepsilon>0$ and let $\gamma^{\varepsilon}$ denote the map defined by (4.14). The following statements hold.

$\gamma^{\varepsilon}$ is the unique mild solution of the integro-PDE (4.1), see Definition 4.1.

$\gamma^{\varepsilon}$ is the unique solution to the regularized Feynman-Kac equation (4.3).

The existence of a mild solution $\gamma^{\varepsilon}$ of (4.1) has already been proved through the discussion just above. It remains to justify uniqueness. Consider $\gamma^{\varepsilon,1}$, $\gamma^{\varepsilon,2}$ be two mild solutions of (4.1). Then, with similar computations as the ones leading to inequality (4.9), there exists a constant $\mathfrak{C}:=\mathfrak{C}(M_{\Lambda},L_{\Lambda},\|K_{\varepsilon}\|_{\infty},\|\nabla K_{\varepsilon}\|_{\infty})>0$ such that

for all $\lambda>0$ and where we recall that $\|\cdot\|_{TV,\lambda}$ has been defined by (4.6). Taking $\lambda>\mathfrak{C}$, uniqueness follows. This shows item 1. Item 2. follows then by Proposition 4.2. ∎

The theorem below states the convergence of the solution of the regularized Feynman-Kac equation (4.3) to the solution to the Feynman-Kac equation (3.15). This is equivalent to the convergence of the solution of the regularized PDE (4.1) to solution of the target PDE (1.4), when the regularization parameter $\varepsilon$ goes to zero.

Suppose the validity of Assumption 2. For any $\varepsilon>0$, consider the real valued function $u^{\varepsilon}$ such that for any $t\in[0,T]$,

where $\gamma^{\varepsilon}$ is the unique solution of (4.3) (or equivalently the unique mild solution of (4.1)). Then $u^{\varepsilon}$ converges to $u$, the unique solution of (3.15) (or equivalently the unique mild solution of (1.4)). More precisely we have

Before proving Theorem 4.5, we state and prove a preliminary lemma.

Suppose the validity of Assumption 2. Consider $u$ the unique solution of (3.15), then for all $t\in[0,T]$

In this proof, $C$ denotes a real constant that may change from line to line, only depending on $M_{\Lambda}$, $L_{\Lambda}$, $C_{u}$ and $\|u_{0}\|_{\infty}$, where we recall that the constant $C_{u}$ only depends on $\Phi,g$ and come from inequality (6.15).

By integrating the absolute value of both sides of (4) w.r.t. $dx$, it follows there exists a constant $C>0$ such that

is the law density of $Y_{s}$, by inequality (6.16) of Lemma 6.4 we get

Injecting (4.26) and (4.27) into the r.h.s. of (4.24), it comes

These equalities follow by computing the derivative of $u(t,\cdot)$ and $u^{\varepsilon}(t,\cdot)$ in the sense of distributions. Taking into account (4.29) and (4.30), it is easy to see that very similar arguments as those invoked above to prove (4.28), lead to

Gathering (4.28) together with (4.31) yields

Applying Gronwall’s lemma to the real-valued function

Moreover, by inequalities (6.14) and (6.15) of Lemma 6.4, for $0\leq s<t\leq T$,

Lebesgue dominated convergence theorem then implies that the third and fourth terms in the r.h.s. of (4.33) converge to when $\varepsilon$ goes to . This ends the proof. ∎

We assume here that $K$ verifies (2.6). Let $u^{\varepsilon}$ be the real-valued function defined by (4.16). Under Assumption 2 and in the particular case where the function $(t,x,y,z)\mapsto\Lambda(t,x,y,z)$ does not depend on the $z$ variable corresponding to the gradient $\nabla u$, there exists a constant

with $C_{u}$ denoting the constant given by (6.14) (only depending on $\Phi$, $g$) such that the following holds. For all $t\in]0,T]$,

In the proof $C$ is a constant fulfilling (4.37). The arguments are the same as the ones used in the proof of Theorem 4.5 since in the present case, $\Lambda$ only depends on $(t,x,u)$ and not on $\nabla u$. In particular, we obtain for $t\in]0,T]$,

Particles system algorithm

To simplify notations in the rest of the paper, $f_{t}$ will denote $f(t)$ where $f:[0,T]\rightarrow E$ is an $E$-valued Borel function and $(E,d_{E})$ a metric space.

where $(K_{\varepsilon})_{\varepsilon>0})$ are mollifiers fulfilling (2.4) and (2.5). We recall that $V_{t}$ is given by (2.1). The first line of (5.1) is a $d$-dimensional classical SDE whose strong existence and pathwise uniqueness are ensured by classical theorems for Lipschitz coefficients. Moreover $\xi^{i},i=1,\cdots,N$ are i.i.d. In the following lemma, we prove by a fixed-point argument that the third line equation of (5.1) has a unique solution.

From now on, it remains to ensure that $T^{\varepsilon,N}$ is indeed a contraction with respect $\|\gamma\|_{TV,\lambda}$ for some $\lambda$. To simplify notations, we set for all $i\in\{1,\cdots,N\}$,

with $C=C(T,\|K_{\varepsilon}\|_{\infty},\|\nabla K_{\varepsilon}\|_{\infty},L_{\Lambda},M_{\Lambda})$. It follows that

By taking $\lambda>C$ and invoking Banach fixed-point theorem, we end the proof. ∎

After the preceding preliminary considerations, we can state and prove the main result of the section.

We suppose the validity of Assumption 2. Assume that the kernel $K$ is verifying (2.7). Let $u^{\varepsilon}$ be the real valued function defined by (4.16), and $u^{\varepsilon,N}$ such that for any $t\in[0,T]$,

where $\gamma^{\varepsilon,N}$ is defined by the third line of (5.1). There is a constant $C$ (only depending on $M_{\Phi}$, $M_{g}$, $M_{\Lambda}$, $\|K\|_{\infty}$, $\|\nabla K\|_{\infty}$, $L_{\Phi}$, $L_{g}$, $L_{\Lambda}$, $T$, $\|u_{0}\|_{\infty}$ and $C_{u}$) such that the following holds.

In the particular case where the function $(t,x,y,z)\mapsto\Lambda(t,x,y,z)$ does not depend on the $z$ variable (corresponding to the gradient $\nabla u$), then previous item holds replacing (5.9) with

One can first decompose the error on the l.h.s of inequality (5.9) as follows

By the boundedness assumption on $\Lambda$ (item 5. of Assumption 2.),

Proceeding similarly for the term $B^{\prime\varepsilon,N}_{t}$ leads to

We are now interested in bounding the r.h.s. of (5.21).

Recalling (5.1), (5.12) and inequality (2.3), we have

By inequality (6.16) in Lemma 6.4 and (4.25), $\|p_{s}\|_{\infty}\leq C_{u}\|u_{0}\|_{\infty}$ for all $s\in[0,T]$. Recalling that $\gamma^{\varepsilon}$ verifies (4.3), using inequality (5.18), we obtain

Moreover, for all $s\in[0,T]$, the boundedness of $|K|$ and $|\nabla K|$ implies

Now let us treat the proof of (5.10), in the specific case where $\Lambda$ does not depend on $\nabla u$. Adapting (5.1) when $\nabla u$ does not appear in $\Lambda$ yields

Considering an additional particle $\xi^{0}$ such that $(\xi^{0},\xi^{1},\cdots,\xi^{N})$ are i.i.d. yields

Injecting the above inequality in (5.28) and using triangle inequality yields (reminding that $C$ is a constant that may change from line to line)

Using the fact that $\|p_{s}\|_{\infty}\leq C_{u}\|u_{0}\|_{\infty}$ by (6.16) and inequality (5.18), implies that for $\varepsilon$ small enough we obtain

We suppose the validity of Assumption 2. Let Assume that the kernel $K$ is verifying (2.6) and (2.7).

If $\varepsilon\rightarrow 0$, $N\rightarrow+\infty$ such that $\frac{1}{\sqrt{N\varepsilon^{d+4}}}e^{\frac{C}{\varepsilon^{d+1}}}\rightarrow 0\ ,$ (where $C$ is the constant coming from Proposition 5.2) then

In the particular case where the function $(t,x,y,z)\mapsto\Lambda(t,x,y,z)$ does not depend on the $z$ variable (corresponding to the gradient $\nabla u$), there is a constant $C$ (only depending on $\kappa$, $C_{u}$, $M_{\Phi}$, $M_{g}$, $M_{\Lambda}$, $\|K\|_{\infty}$, $\|\nabla K\|_{\infty}$, $L_{\Phi}$, $L_{g}$, $L_{\Lambda}$, $T$, $\|u_{0}\|_{\infty}$), such that the following holds.

where we have used Proposition 5.2 for the second inequality above. Taking into account Theorem 4.5 above, it appears clearly that the convergence of $u^{\varepsilon,N}$ (resp. $\nabla u^{\varepsilon,N}$) to $u$ (resp. $\nabla u$) will hold as soon as $\frac{1}{\sqrt{N\varepsilon^{d+4}}}e^{\frac{C}{\varepsilon^{d+1}}}\rightarrow 0$ when $\varepsilon\rightarrow 0$, $N\rightarrow+\infty$. This concludes the proof of (5.32).

The second inequality (5.33), concerning the specific case where $\Lambda$ does not depend on the gradient $\nabla u$), is proved similarly by gathering inequality (4.38) from Proposition 4.7 and inequality (5.10) from Proposition 5.2. ∎

In the first statement of Corollary 5.3 appears the condition $\frac{1}{\sqrt{N\varepsilon^{d+4}}}e^{\frac{C}{\varepsilon^{d+1}}}\rightarrow 0$ when $\varepsilon\rightarrow 0$, $N\rightarrow+\infty$. This requires a "trade-off" between the speed of convergence of $N$ and $\varepsilon$. Setting $\Phi(\varepsilon):=\varepsilon^{-(d+4)}e^{\frac{2C}{\varepsilon^{d+1}}}$, the trade-off condition can be formulated as

An example of such trade-off between $N$ and $\varepsilon$ can be given by the relation $\varepsilon(N)=\left(\frac{1}{\log(N)}\right)^{\frac{1}{d+4}}$.

The estimate (5.33) recovers the same order of convergence as the one encountered in classical density estimates, see e.g. (22) in . This happens in spite of the fact the weights $V_{t}$ in (5.1) depend on the whole past of the whole particle system.

Appendix

We apply (16) in Lemma 7 (p.251) of setting $g=\sqrt{f},\ \varepsilon=1$. ∎

From Lemma 6.1, we deduce the following lemma.

Then for any strictly positive real $\varepsilon\leq(1/I(G))^{\frac{1}{d+1}}$,

and $G_{\varepsilon}(\cdot):=\frac{1}{\varepsilon^{d}}G(\frac{\cdot}{\varepsilon})$.

Hence, as soon as $\varepsilon\leq(1/I(G))^{\frac{1}{d+1}}$, we have

which, owing to (6.4), concludes the proof. ∎

where $x_{i}$ is the $i$-th coordinate of $x$ and $H_{t}$ given by (6.7). Observe that, for any $\varepsilon>0$, $1\leq i\leq d$,

Taking the $L^{p}$ norm in equality (6.11), Young’s inequality yields

which gives the result by recalling (6.10). ∎

2 About transition kernels

In the following lemma, we state well-known technical properties about the transition probability function of a diffusion process. All the statements below are established in .

We assume here the validity of items 1. to 3. of Assumption 2. Consider a stochastic process $Y$, solution of the SDE

where $q(s,x_{0},t,x):=\left(\frac{c_{u}(t-s)}{\pi}\right)^{\frac{d}{2}}e^{-c_{u}\frac{|x-x_{0}|^{2}}{t-s}}$ is a Gaussian kernel.

In particular for all $t\in[0,T]$ for almost all $r,x$ we have

See Theorem 5.4, Section 5 in Chapter 6 in , Section 4 of , the fact that classical solutions of (2.15) are also distributional solutions together with Theorem 15, Section 9, chap. 1 in and inequalities (8.13) and (8.14) just before. ∎

3 Proof of Proposition 2.2

For given $u$ we set $\hat{\Lambda}(s,x):=\Lambda(s,u(s,x),\nabla u(s,x))u(s,x)$. We first suppose that $u$ is a mild solution of (1.4). Taking into account that $P(s,x_{0},t,\cdot)$ is a distributional solution of (2.18), we show below that $u$ is indeed a weak solution of $\eqref{eq:PDE}$.

Indeed, for $0\leq r<t\leq T$, (2.16), gives

For every $s\in[0,T]$ we define the measure

Since $u$ is a mild solution of (1.4), see (2), we have

Starting with the left-hand side of (6.19), we start by plugging in the expression of $v$ in (6.18) into the right-hand side of (6.19):

For the first term on the right-hand side of (6.20), we can directly use identity (6.17) to infer

For the second term on the right-hand side, a simple application of Fubini’s Theorem for random kernels enables us to “pull out” the integral with respect to $r$ and then apply (6.17):

Plugging the equalities (6.21), (6.22), into (6.20) leaves us with

where for the latter equality we again used the definition of $v$ in (6.18). This is exactly (6.19) and thus completes the first part of the proof.

Conversely, suppose that $u$ is a weak solution of (1.4), in the sense of Definition 2.1. We also consider

We want to ensure that $u=\bar{v}$. On the one hand, by the first part of the proof applied to $\bar{v}$ instead of $v$ we can show that

On the other hand, $u$ being a weak solution of (1.4), it also a solution of (6.24) in the sense of distributions. We set $w:=\bar{v}-u$. It follows that $w$ and the zero measure function $\bar{w}\equiv 0$ $\hat{v}:=0$ both satisfy (2.15) in the sense of distributions, see (2.16). Uniqueness of the solution of (2.15) implies that $w=0$, which concludes the proof.

4 Proof of technicalities of Section 3

We give in this section the proof of Lemma 3.4.

We only prove the direct implication since the converse follows easier with similar arguments. The aim is to prove, for all $n\in\{1,\cdots,N\}$,

We are going to proceed by induction on $n$. For $n=1$, formula (6.25) follows from (3.13) by taking $k=0$. We suppose now that $(H_{n-1})$ holds for some integer $n\geq 1$. Then, by taking $t=(n-1)\tau$ in the first line equation of (6.25), it follows immediately that

On the other hand, since (3.13) is valid for all $t\in[(n-1)\tau,n\tau]$ by plugging $k=n-1$, we obtain

for all $t\in[(n-1)\tau,n\tau]$. Inserting (6.4) in (6.4) yields

Invoking the Chapman-Kolmogorov equation satisfied by the transition probability function $P(s,x_{0},t,dx)$ (see e.g. expression (2.1) in Section 2.2, Chapter 2 in ), we have

Applying (6.29) with $\theta=(n-1)\tau$, it follows that for all $t\in[0,n\tau]$,

ACKNOWLEDGMENTS. The authors are grateful to the Editors and to the two Referees who have read extremely carefully the paper stimulating with useful observations its improvement.

References