Stochastic models for fully coupled systems of nonlinear parabolic equations

TL;DR

This paper develops a probabilistic framework for representing solutions to complex coupled nonlinear parabolic PDE systems modeling interacting populations, enabling new stochastic analysis tools.

Contribution

It introduces a novel probabilistic representation for fully coupled nonlinear parabolic systems using stochastic equations and Markov processes.

Findings

Probabilistic representation of coupled PDE systems established.

Stochastic equations derived for nonlinear Markov processes.

Framework applicable to models of spatial population segregation.

Abstract

We construct a probabilistic representation of a system of fully coupled parabolic equations arising as a model describing spatial segregation of interacting population species. We derive a closed system of stochastic equations such that its solution allows to obtain a probabilistic representation of a weak solution of the Cauchy problem for the PDE system. The corresponded stochastic system is presented in the form of a system of stochastic equations describing nonlinear Markov processes and their multiplicative functionals.

Equations

$${\mathbf{u}}_t^q=\frac{1}{2}\sum _{i,j=1}^d{F}_u^{ij}(x){\nabla }_{ij}^2{u}^q+\sum _{i=1}^d{a}_u^i(x){\nabla }_i{u}^q+\sum _{m=1}^{d_1}\sum _{i=1}^d[B_u{]}_i^{mq}(x){\nabla }_i{u}^m+\sum _{m=1}^{d_1}{c}_u^{mq}(x)u^m,$$

$${\mathbf{u}}_t^q=div{G}^q+f^q(u),l=1,2,$$

$$u_t^1=\Delta (u^1(d_1+d_{11}{u}^1+d_{12}{u}^2))+u^1({\alpha }_1-{\alpha }_{11}{u}^1-{\alpha }_{12}{u}^2),u^1(0,x)=u_0^1(x),$$

$$u_t^2=\Delta (u^2(d_2+d_{21}{u}^1+d_{22}{u}^2))+u^2({\alpha }_2-{\alpha }_{21}{u}^1-{\alpha }_{22}{u}^2),u^2(0,x)=u_0^2(x),$$

$$\Vert u{\Vert }_k={\left({\int }_{R^d}[|u(x){|}^2+\sum _{|\alpha |\le k}\Vert {\nabla }^{\alpha }u(x){\Vert }^2]dx\right)}^{\frac{1}{2}},$$

$$⟨⟨u,h⟩⟩={\int }_{R}u(x)\cdot h(x)dx,$$

$$\Vert u{\Vert }_{\mathcal{W}}^2={\int }_{R^d}{\int }_0^T[\Vert u(t){\Vert }_{L^2}^2+\Vert \nabla u(t){\Vert }_{L^2}^2]dt.$$

$${\partial }_s{\int }_{R^d}{u}^q(s,x)h(x)dx={\int }_{R^d}{u}^q(s,x)[M_u^q\Delta h(x)+c_{u^1,u^2}^q(x)]h(x)dx$$

$$M_u^q=d_q+d_{q1}{u}^1+d_{q2}{u}^2,c_u^q={\alpha }_q-{\alpha }_{q1}{u}^1-{\alpha }_{q2}{u}^2.$$

$${\int }_0^t⟨u^q(s),\left\{{\partial }_{s}h+\frac{1}{2}[M_u^q{]}^2\Delta h(s)+c_{u^1,u^2}^q(s)h(s)⟩\right\}ds$$

$$=⟨u^q(t),h(t)⟩-⟨u^q(0),h(0)⟩$$

$$M_u^q=\sqrt{2[d_q+d_{q1}{u}^1+d_{q2}{u}^2]},c_u^q={\alpha }_q-{\alpha }_{q1}{u}^1-{\alpha }_{q2}{u}^2.$$

$${\int }_0^T{\int }_{R^d}{u}^q(t,x)[{\partial }_t(\rho h)+{\mathcal{M}}^q(\rho h)](t,x)dxdt=0,$$

$${\mathcal{M}}^{q}h(t,x)=M_u^q\Delta h(t)+c_u^q(t)h.$$

$$-{\int }_0^T\frac{d}{dt}\rho (t){\int }_{R^d}{u}^q(t,x)h(t,x)dxdt={\int }_0^T\rho (t){\int }_{R^d}{u}^q(t,x)[{\partial }_{t}h(t,x)+\mathcal{L}h(t,x)]dxdt.$$

$$\frac{d}{dt}{\int }_{R^d}{u}^q(t,x)h(t,x)dx={\int }_{R^d}{u}^q(t,x)[{\partial }_{t}h(t,x)+{\mathcal{M}}_u^{q}h(t,x)]dx.$$

$${\int }_{R^d}{u}^q(t,x)h(t,x)dx=C+{\int }_0^t{\int }_{R^d}{u}^q(\theta ,x)[{\partial }_{\theta }h(\theta ,x)+{\mathcal{M}}_u^{q}h(\theta ,x)dxd\theta \text{mbox}fora.a.t\in [0,T].$$

$$\underset{t\to 0}{\lim }{\int }_{R^d}{u}^q(t,x)h(0,x)dx\to {\int }_{R^d}{u}_0^q(x)h(0,x)dx$$

$${\partial }_s{h}^q+M_u^q(x)\Delta h^q+c_u^q(x)h^q=0,h^q(T,x)=h^q(x),0\le s\le t\le T,$$

$$d{\xi }^q(\theta )=M_u^q({\xi }^q(\theta ))dw(\theta ),{\xi }^q(s)=y,0\le s\le \theta \le t\le T$$

$$d{\eta }^q(\theta )=c_u^q({\xi }^q(\theta )){\eta }^q(\theta )d\theta ,{\eta }^q(s)=1,q=1,2.$$

$$h^q(s,y)=E[{\eta }^q(t)h^q({\xi }_{s,y}^q(t))].$$

$$d{\eta }^q(\theta )={\tilde{c}}_u^q(\xi (\theta )){\eta }^q(\theta )d\theta +C_u^q(\xi (\theta )){\eta }^q(\theta )dw(\theta ),{\eta }^q(s)=1,q=1,2,$$

$$u^q(s,x)=E_{s,x}\left[{\hat{\eta }}^q(t)u_0^q({\hat{\xi }}^q(t))\right],q=1,2,$$

$$⟨u^q(t),h⟩=⟨u^q(0),h⟩-{\int }_0^t⟨[\Delta [u^q(\theta )M_u^q(\theta )]+c_u^q(x)u^q(\theta )]h⟩\}d\theta$$

$$d{\hat{\xi }}_{0,x}^q(\theta )=[M_u^q\nabla M_u^q]({\hat{\xi }}_{0,x}^q(\theta ))d\theta +M_u^q({\hat{\xi }}_{0,x}^q(\theta ))d\tilde{w}(\theta ),{\hat{\xi }}_{0,x}^q(0)=x.$$

$$d{\mathbf{J}}^q(\theta )=\nabla M_u^q(\theta ){\mathbf{J}}^q(\theta )dw(\theta ),{\mathbf{J}}^q(s)=I,$$

$$J_{0,t}^q(\omega )>0\text{mbox}and{J}_{s,s}^q(\omega ))=1.$$

$$d{J}^q(\theta )=J^q(\theta )Tr[{\mathbf{J}}^q(\theta )d{\mathbf{J}}^q(\theta )],J^q(s)=1.$$

$$d{\psi }_{0,x}^q(\theta )={\lambda }_u^q({\psi }_{0,x}^q(\theta ))d\theta +{\Lambda }_u^q({\psi }_{0,x}^q(\theta ))dw(\theta ),{\psi }_{0,x}^q(0)=x,$$

Full text

1 Introduction

Systems of nonlinear parabolic equations arise as models in description of various physical, chemical and biological phenomena. To mention some of them recall such phenomena as chemotaxis which is a biological phenomenon describing the change of motion of a population densities or of single particles in response (taxis) to an external chemical stimulus spread in the environment [18]. Another example, are cross-diffusion systems which describe segregation processes in dynamics of interacting populations [12], [20] and in general a number of diffusive conservation laws. Mathematical models of these conservation laws are presented as systems of quasilinear parabolic equations in both divergent and non divergent form.

Actually, from the probabilistic point of view systems of nonlinear parabolic equations can be divided into several large classes.Let us mention here two of them, namely, systems of type 1 [17] having diagonal entries of second order terms

[TABLE]

where coefficients $a_{u},B_{u},c_{u}$ and $F^{ij}_{u}=\sum_{k=1}^{d}A^{ki}_{u}A_{u}^{jk},$ have the form $a_{u}(x)=a(x,u(x),\nabla u(x))$ and systems of type 2 [1] with nondiagonal entries of second order terms such as

[TABLE]

where $G^{q}=\alpha^{q}\nabla u^{q}+\beta^{q}\nabla(u^{q})^{2}+\gamma^{q}\nabla[u^{1}u^{2}]$ and $\alpha_{l},\beta_{l},\gamma_{l}$ are constants.

The results concerning weak and classical solutions of the Cauchy problem for type 1 parabolic systems could be found in the monograph [17] and a number of more recent papers. Probabilistic approach to investigation of different types of the Cauchy problem solution (namely, classical, generalized and viscosity) for nonlinear parabolic systems of type 1 was developed for many years. In particular, probabilistic representations for classical solutions of the Cauchy problem for nonlinear parabolic systems of this kind were constructed in [5] -[BD2], for generalized solutions in [8] and for viscosity solutions in [3].

Investigation of parabolic systems of type 2 started by Amann [1] was developed by many researches especially in connections with problems arising in applications. These systems are sometimes called cross-diffusion systems nevertheless there are only few papers devoted to the investigation of their stochastic origin [10], [11], [13] or the stochastic representation of their solutions [4].

This paper is aimed to develop ideas stated in [3], [4].

We consider a particular case of the above system with cross diffusion used to model the segregation of interacting species proposed in [20]. The system under consideration has the form

[TABLE]

[TABLE]

where $u^{1}$ and $u^{2}$ are the densities of two competing species.

Here and below we use notations $u_{t}=\frac{\partial u}{\partial t}$ for a partial derivative of $u(t,x)$ in $t\in[0,T]$ and $\nabla u=(\frac{\partial u}{\partial x_{1}},\dots,\frac{\partial u}{\partial x_{d}})$ for the gradient of $u$.

Note that a probabilistic approach to investigation of the parabolic equations includes there steps. The first is under assumption that there exists a unique solution of the PDE problem (as a rule rather regular) one have to derive a potential probabilistic representation in terms of some diffusion processes. The second step is to derive a closed stochastic system for these diffusion processes and the third step is to investigate the stochastic problem and construct its solution with the required properties which allow to check that it gives rise to the solution of the original PDE problem.

In our previous papers this program was partly realized. The aim of this paper is to derive a probabilistic interpretation of a generalized solution of the Cauchy problem for this system and to construct a closed stochastic system of stochastic equations.

A probabilistic approach to construct a representation of a generalized solution to the Cauchy problem for a scalar nonlinear parabolic equation based on the results of the Kunita stochastic flows theory [14]–[16] was developed in [7], [8].

The main problem we meet while extending this approach to systems with cross diffusion is the fact that we cannot apply directly the Ito formula which is a main tool in construction of probabilistic representation of a generalized solution of the Cauchy problem for scalar nonlinear parabolic equations [7] and systems of nonlinear parabolic equations with diagonal principal part [8].

Nevertheless using the stochastic flow approach we succeed to derive the probabilistic representation of a generalized solution to (1.3), (1.4).

2 Probabilistic models for a fully coupled system of PDEs

2.1 Dual PDEs

To define a generalized solution of the Cauchy problem (1.1), (1.2) we have to introduce a number of functional spaces.

$\bullet$ The set $C^{k}_{b}(R^{d};R^{d_{1}})$ of $C^{k}$-smooth functions $f:R^{d}\to R^{d_{1}}$ whose partial derivatives of order less or equal then $k$ are bounded.

$\bullet$ The set $C^{1,k}_{0}([0,T]\times R^{d},R^{d_{1}})$ of continuous functions $u:[0,T]\times R^{d}\to R^{d_{1}}$ with compact supports with continuous first partial derivatives in $t$ and partial derivatives of order less or equal then $k$ in $x\in R^{d}$.

$\bullet$ The set $L^{2}=L^{2}(R^{d};R)$ of real valued functions with the norm $\|u\|_{L^{2}}=\left(\int_{R^{d}}|u(x)|^{2}dx\right)^{\frac{1}{2}}$ which is the Hilbert space with the inner product $\langle u,v\rangle=\int_{R^{d}}u(x)v(x)dx$ for $u,v\in L^{2}$.

$\bullet$ The Sobolev space $H^{k}=\{u\in L^{2}:\nabla^{k}u\in L^{2}\}$, where $\nabla^{k}u(x)$ is a generalized $k$-th order derivative with the norm

[TABLE]

where $|\alpha|=\sum\alpha_{j}$.

$\bullet$ The Schwartz space ${\cal D}=C^{\infty}_{0}(R^{d})$ and the space $H^{k}_{0}=H^{k}_{0}$ which is a completion of $\cal D$ in the norm $\|\cdot\|_{k}.$ Given $R^{2}$ -valued functions $u,h$ with components in ${\cal D}$ set

[TABLE]

where $u\cdot h=u_{1}h_{1}+u_{2}h_{2}.$

$\bullet$ ${\cal W}_{T}=\{u:u\in L^{2}((0,T);L^{2}(R^{d}))\cap L^{2}((0,T);H^{1}_{0}(R^{d}))\}$ – the set of functions $t\mapsto u(t,\cdot)$ from $(0,T)$ to $L^{2}(R^{d})$ with the norm

[TABLE]

A couple of functions $u^{1},u^{2}\in{\cal W}_{T}$ is a weak (generalized) solution of the Cauchy problem (1.3), (1.4) provided $u^{1},u^{2}\in{\cal W}_{T}$ and equalities

[TABLE]

hold for arbitrary $h\in C_{0}^{1,\infty}([0,T]\times R^{d})$ and $q=1,2$. Here

[TABLE]

Along with this definition we need one more which gives a prompt concerning to what sort of diffusion processes we should be interested in.

We say that a couple of functions $u^{1},u^{2}\in{\cal W}_{T}$ is a weak (generalized) solution of the Cauchy problem (1.3), (1.4) provided $u^{1},u^{2}\in{\cal W}_{T}$ and equalities

[TABLE]

[TABLE]

hold for arbitrary $h\in C_{0}^{1,\infty}([0,T]\times R^{d})$ and $q=1,2$. Here

[TABLE]

We say that a couple $(u^{1},u^{2})\in{\cal W}^{2}_{s,T}$ is a regular solution of the Cauchy problem (1.3)–(1.4) provided $(u^{1},u^{2})\in C^{1,2}([0,T]\times R^{d};R^{2})\cap{\cal W}^{2}_{T}$ in addition to the above integrals identities.

It should be noted that similar to [9] we can prove the following statement.

Lemma 2.1. *Let $u^{q}(t,x),q=1,2$ be a solution to (1.3)–(1.4) such that $\sup_{\theta\in(0,T)}|u^{q}(t,x)|<\infty$ and $u^{q}\in L^{1}([0,T]\times G)$ for each compact set $G\in R^{d}$. Then (2.3) holds for a.a. $t\in[0,T]$ and for every function $h\in C_{b}([0,T]\times R^{d})\cap C_{b}^{1,2}((0,T)\times R^{d})$ *

Proof. It is enough to check this equality for any $h(t,x)=0$ for $\|x\|>R$ for some $R>0$. Let $\rho\in C^{\infty}_{0}((0,T))$. Since $u^{q}$ satisfies (1.3)–(1.4), then applying integration by part formula we easily check that

[TABLE]

where

[TABLE]

Hence

[TABLE]

This yields that the function $t\mapsto\int_{R^{d}}u^{q}(t,x)h(t,x)dx$ has an absolutely continuous version and

[TABLE]

Finally for some real number $C$

[TABLE]

Since by assumption $h(t,x)$ converges uniformly to $h(0,x)$ as $t\to 0$ we get

[TABLE]

and thus $C=\int_{R^{d}}h(0,x)u^{q}_{0}(x)dx.$

Assume that there exists a unique $C^{2}$-regular generalized solution $(u^{1},u^{2})$ to (1.3)-(1.4), such that $\inf_{x}u^{1}(t,x)\geq 0,\inf_{x}u^{2}(t,x)\geq 0$.

To construct stochastic processes associated with the Cauchy problem

[TABLE]

we consider a stochastic differential equation (SDE)

[TABLE]

and a linear SDE

[TABLE]

Here $w(t)$ is a standard $R^{d}$-valued Wiener processes defined on a given probability space $(\Omega,{\cal F},P).$ By the above assumption coefficients $M^{q}_{u}$ and $c^{q}_{u}$ in (2.7), (2.8) are bounded Lipschitz continuous nonrandom functions. Hence, by standard reasoning we can prove that there exists a Markov process $\xi^{q}(t)$ satisfying (2.7) and a multiplicative functional $\eta^{q}(t)$ (of the process $\xi^{q}(t)$) which satisfies (2.8). In addition for $h\in C^{2}(R^{d})\cap H^{2}(R^{d})$ there exists a unique classical solution $h^{q}$ of the Cauchy problem (2.6) that admits a probabilistic representation in the form

[TABLE]

Later we will see that to deal with the original Cauchy problem (1.3)–(1.4) we have to consider a more complicated linear SDE of the form

[TABLE]

with coefficients $\tilde{c}^{q}_{u}=c^{q}_{u}+\|\nabla M_{u}\|^{2}$ and $C_{u}=-\nabla M_{u}.$

Along with the process $\xi^{q}(\theta)$ we will need the process $\hat{\xi}^{q}(\theta)=\xi^{q}(t-\theta)$. One can easily see that $\xi^{q}(t)=\hat{\xi}^{q}(0)$ and $\hat{\xi}^{q}(t)=\xi^{q}(0)$.

The process $\xi^{q}(\theta)$ satisfying (2.7) and its time reversed $\hat{\xi}^{q}(\theta)=\xi^{q}(t-\theta)$ play an important role in construction of probabilistic representations for generalized solutions of (1.3)–(1.4). Similar to [4] based on the results of [15],[16] we can prove the following statement.

Theorem 2.2 Given strictly positive regular (weak) solution $u^{1},u^{2}\in{\cal W}_{0,T}\cap C^{1,2}$ of the Cauchy problem (1.3)–(1.4) let $\xi^{q}(\theta),\eta^{q}(\theta),\,q=1,2$ satisfy (2.7), (2.10) while $\hat{\xi}^{q}(\theta)$ and $\hat{\eta}^{q}(\theta)=\eta^{q}(t)\circ\psi^{q}_{\theta,t}$ be the corresponding time reversal processes.

Then functions

[TABLE]

satisfy the integral identities

[TABLE]

Note that the system (2.7) ,(2.10) ,(2.11) is not closed since coefficients of SDEs governed the processes $\hat{\xi}^{q}(\theta)$ and $\hat{\eta}^{q}(\theta)$ depend on $u^{q}$ and $\nabla u^{q}$. Hence to make this system closed we need probabilistic representations both for functions $u^{q}$ and their gradients $\nabla u^{q}$ as well.

2.2 Stochastic flows

We start with introducing some required functional spaces.

Given a Wiener process $w(\theta)$ denote by $\tilde{w}(\theta)=w(t-\theta)-w(t)$ for a fixed $t>\theta$ and given a stochastic process $\xi^{q}(\theta),s\leq\theta\leq t$ denote by $\hat{\xi}^{q}(\theta)=\xi^{q}(t-\theta)$ its time reversal process. Let $\phi^{q}_{0,\theta}:R^{d}\to R^{d}$, $\psi^{q}_{0,\theta}:R\to R^{d}$ be stochastic flows generated by $\xi^{q}(\theta)$ and $\hat{\xi}^{q}(\theta)$ that is $\xi^{q}_{0,y}(\theta)=\phi^{q}_{0,\theta}(y),\quad\hat{\xi}^{q}_{0,x}(\theta)=\psi^{q}_{0,\theta}(x)$, that is $\phi^{q}_{0,t}(\psi^{q}_{0,t}(x))=x$.

Given bounded functions $u^{q}(\theta,x)\in R^{2},q=1,2,$ defined on $[0,T]\times R^{d}$ and differentiable in $x$ we consider a stochastic equation (2.7). One can check (see [19], [6]) that the time reversal process $\hat{\xi}_{q}^{0,x}(\theta)$ satisfies the equation

[TABLE]

where, $0\leq\theta\leq t$.

In addition under the above assumptions the mappings $\phi^{q}_{0,\theta}:x\mapsto\xi^{q}_{0,\theta}(x)$ and $\psi^{q}_{0,\theta}:y\mapsto\xi^{q}_{0,\theta}(y)$ are differentiable. The process ${\bf J}^{q}(t)=\nabla\xi^{q}(t)$ satisfies the SDE

[TABLE]

where $I$ is the identity matrix. Denote by $J^{q}_{0,t}(\omega)={\rm det}{\bf J}^{q}_{0,t}(\omega)$ and note that

[TABLE]

Moreover $J^{q}_{0,t}$ satisfies the SDE

[TABLE]

The relation (2.15) can be easily deduced from a known formula for a differential of a determinant of a matrix and polylinearity of the function $det{\bf J}^{q}$ w.r.t rows of the matrix ${\bf J}^{q}$. Note that for a linear function $f(x)$ we have $df(\xi(t))=\nabla f\cdot d\xi(t)$ that is there are no correction terms in the Ito formula.

Along with SDE (2.13) we will need below an alternative SDE for the process $\hat{\xi}^{q}(\theta)$. To derive the required SDE we apply the Ito-Wentzell formula to the composition $\phi^{q}_{0,t}\circ\psi^{q}_{0,t}$. Let

[TABLE]

where coefficients $\Lambda^{q}_{u}$ and $\lambda^{q}_{u}$ should be defined to ensure that $\psi^{q}_{0,t}(x)$ is time reversal to $\phi^{q}_{0,t}(\kappa)$. Since in this case $\phi^{q}_{0,t}\circ\psi^{q}_{0,t}(x)=x$ then we get

[TABLE]

[TABLE]

Hence

[TABLE]

[TABLE]

Thus the process $\hat{\xi}^{q}(t)\equiv\hat{\xi}^{q}_{0,x}(t)=\psi^{q}_{0,t}(x)$ satisfies the SDE

[TABLE]

As a result the following assertion holds.

Theorem 2.3 *Let $\phi^{q}_{s,t}(\kappa)$ be a solution of (2.12) with $C^{k}$ -smooth coefficients for $(k\geq 3)$. Then the inverse flow $[\phi^{q}_{s,t}]^{-1}=\psi^{q}_{s,t}$ satisfies (2.16) and is a diffeomorphism. *

Given a generalized function $u^{q}_{0}$ we define a composition of $u^{q}_{0}$ with a stochastic flow $\psi^{q}_{s,t}(\omega)$ as a random variable valued in the space ${\cal D}^{\prime}$ dual to ${\cal D}$. Note that given $h\in{\cal D}$ a product $h\circ\phi^{q}_{s,t}(\omega)J^{q}_{s,t}(\omega)$ belongs to $\cal D$. Set

[TABLE]

One can check that (2.17) defines a linear functional on ${\cal D}$ which we denote by $u^{q}_{0}\circ\psi^{u}_{s,t}$. If $m$ can be represented as $u^{q}_{0}=u^{q}_{0}(x)dx,$ where $u^{q}_{0}(x)$ is a continuous function then $u^{q}_{0}\circ\psi^{u}_{s,t}$ is just a composition of $u_{0}$ with $\psi^{u}_{s,t}$ due to an equality

[TABLE]

resulting from the changing variable formula under the integral sign

[TABLE]

By similar arguments we conclude that when $u^{q}(t)\circ\psi^{q}_{0,t}(x)=\hat{\eta}^{q}_{0,x}(t)u_{0}^{q}(\hat{\xi}^{q}_{0,x}(t))$,

[TABLE]

[TABLE]

Let

[TABLE]

and ${\cal M}^{q}_{u}$ be dual operators to ${\cal L}^{q}_{u}$ in $L^{2}(R^{d})$, that is

[TABLE]

By the above considerations we note that given a function $h\in{\cal D}$ one can consider a process $\gamma^{q}(t)$ defined by

[TABLE]

and use $\gamma^{q}(t)$ as stochastic test functions. Actually, since $\langle\hat{\eta}^{q}(t)u^{q}_{0}\circ\psi^{q}_{0,t},h\rangle=\langle u^{q}_{0},\gamma^{q}(t)\rangle$ we can state the following assertion.

**Lemma 2.4.**Let coefficients $\tilde{c}^{q}_{u}$ and $C^{q}_{u}$ have the form

[TABLE]

Then the processes $\gamma^{q}(\theta)=\eta^{q}(\theta)h(\xi^{q}_{0,y}(\theta))J^{q}(\theta),q=1,2,$ have stochastic differentials of the form

[TABLE]

[TABLE]

Proof. We apply the Ito formula to evaluate $d\gamma^{q}(t)$

[TABLE]

[TABLE]

[TABLE]

Taking into account the expressions for $d\xi^{q}(t),d\eta^{q}(t)$ and $dJ^{q}(t)$ from (2.7), (2.10) and (2.15) we deduce

[TABLE]

[TABLE]

Setting

[TABLE]

and

[TABLE]

we get

[TABLE]

[TABLE]

To conclude this section remind (see [15]) that given functions $m^{q}\in H^{1}$ we can define random variables $\langle\eta^{q}(t)m^{q}\circ\psi^{q}_{0,t},h\rangle$ and prove that it has finite moments for $m^{q}\in H^{1}$ and $h\in H^{1}\cap C^{\infty}_{0}={\cal R}$. Hence $\langle K^{q}_{0,t},h\rangle=E\langle\eta^{q}(t)m^{q}\circ\psi^{q}_{0,t},h\rangle$ is a continuous linear functional on $\cal R$ that yields $K^{q}_{s,t}\in H^{1}$ and $K^{q}_{s,t}=E[\eta^{q}(t)m^{q}\circ\psi^{q}_{s,t}]$ is called a generalized expectation of $g^{q}(t)=\eta^{q}(t)m^{q}\circ\psi^{q}_{s,t}$. Denote by $U^{q}(t)m^{q}=E[\eta^{q}(t)m^{q}\circ\psi_{0,t}]$. It is a linear map from $H^{1}$ into itself, possessing the semigroup property $U^{q}(t)U^{q}(s)=U^{q}(s+t)$ for any $s,t>0$. IOne can easily verify this fact using stochastic flow properties.

[TABLE]

[TABLE]

[TABLE]

Due to evolution properties of multiplicative operator functionals we can check as well that families $U^{q}_{0,t}m^{q}=E[\eta^{q}(t)m^{q}(t)\circ\psi_{0,t}],$ have the evolution property.

3 Stochastic counterpart of the Cauchy problem for a system with cross-diffusion

Note that to obtain a closed system of stochastic equations which can be treated as a stochastic counterpart of the system

[TABLE]

it is not enough to construct a stochastic representation of the solution to (3.1) itself since coefficients of SDEs for $\hat{\xi}^{q}(\theta)$ and $\hat{\eta}^{q}(\theta)$ derived in the previous section depend on $\nabla u^{q}$. Hence we need stochastic representations for spatial derivatives of $u^{q}(t,x)$. In the previous section it was shown that we can associate with (3.1) a system of stochastic equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

To make the system (3.4)-(3.6) closed we need an extra relation for the function $v^{q}_{i}(t,x)=\nabla_{i}u^{q}(t,x)$, $i=1,\dots,d$, since coefficients of (3.4) and (3.5) depend on $\nabla u^{q}$.

To derive this relation we need some additional speculations based on results from [12]. By formal differentiation of the system

[TABLE]

we get a PDE for $v^{q}_{i}=\nabla_{i}u^{q}$

[TABLE]

$v^{q}_{i}(0,x)=\nabla_{i}u^{q}_{0}(x).$

In a similar way from

[TABLE]

we get a PDE for $g^{q}_{i}=\nabla_{i}h^{q}$

[TABLE]

$g^{q}_{i}(0,y)=\nabla_{i}h^{q}(y).$

In addition note that we can construct a stochastic representation of the solution to (3.9)-(3.10) in the form

[TABLE]

where $G(t,y)=\pmatrix{h^{q}(t,y)\cr\nabla h^{q}(t,y)}$ and stochastic processes $\xi^{q}(\tau)$ and $\beta^{q}_{ik}(\tau)$ satisfy SDEs

[TABLE]

[TABLE]

Here

[TABLE]

where $\delta$ is the Kronecker delta. Thus for $G_{0}(y)=G^{q}(0,y)=\pmatrix{h^{q}(y)\cr\nabla h^{q}(y)}$ we get

[TABLE]

[TABLE]

To deduce the stochastic representation for the function $g^{q}_{j}=\nabla_{j}h^{q}$ we note that given the PDE system (3.9)-(3.10) we can derive its stochastic representation as follows. Let us rewrite the system (3.7),(3.8) in the form

[TABLE]

where

[TABLE]

Consider as well a dual system derived from (3.11) as follows. Integrate over $R^{d}$ a product of (3.11) and vector test functions $(h,g)^{*}$, where $g_{j}=\nabla_{j}h,\,j=1,\dots,d$. As a result we obtain a system of the form

[TABLE]

where

[TABLE]

Here and below we denote by

[TABLE]

In the sequel we take into account the relation $[d_{q1}v^{1}+d_{q2}v^{2}]\Delta h=[d_{q1}v^{1}+d_{q2}v^{2}]divg$ that allows to construct a proper stochastic representation of the backward Cauchy problem

[TABLE]

based on results of [14].

To this end along with (3.2) we consider a stochastic equation of the form

[TABLE]

with respect to the two component process $\beta^{q}(\theta)=\pmatrix{\beta^{q}_{1}(\theta)\cr\beta^{q}_{2}(\theta)}$ with coefficients $\tilde{c}^{q}$ and $C^{q}$ to be chosen below. Let $\bar{\beta}^{q}(\theta)$ maps $\beta^{q}$ to $\beta^{q}(\theta)$, that is

[TABLE]

Define a stochastic test function

[TABLE]

where $J^{q}(\theta)$ is a Jacobian of the stochastic transformation $y\to\xi^{q}_{s,y}(\theta)$. The stochastic differential of the process $\kappa^{q}(\theta)$ has the form $d\kappa^{q}(\theta)=\pmatrix{d\kappa^{q}_{1}(\theta)\cr d\kappa^{q}_{2}(\theta)}$ with

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Let us specify coefficients $\tilde{c}^{q}$ and $C^{q}$. As it was done in the previous section we choose

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Next we choose

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We do not specify for the moment $N^{q}_{1}$ and $N^{q}_{2}$ since they do not take part in the probabilistic representation of $u^{q}$ and $v^{q}$.

Next we proceed as in the previous section. Denote by $\hat{\beta}^{q}(\theta)$ the time reversal of the process $\bar{\beta}^{q}(\theta)$. To get a closed counterpart of the system (1.1) in addition to theorem 1 we state the following assertion.

**Theorem 3.1.**Under assumptions of theorem 1 both the functions $u^{q}(t,x)$ admit stochastic representations (3.10) and functions $v^{q}_{j}=\nabla_{j}u^{q}$ admit stochastic representations

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Proof. To verify the last assertion of the theorem we note that we have the following matrix relations

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At the other hand from (3.15) we deduce

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By the change of variables $\xi_{0,y}(\theta)=x$ applying stochastic Fubini theorem we get

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Hence we derive that the functions

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satisfy integral identities

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which results due to the assumed uniqueness of a solution to (1.1) that

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and hence

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As a result we deduce from the last equality that (3.14) holds and in addition

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Remark. We have proved that under a priori assumption about existence of a unique regular solution of the Cauchy problem (1.1) there exists a stochastic representation of this solution and moreover we derive a closed system of stochastic equations that can be considered separately without any reference to this a priori assumption. To be more precise we have shown that the system (3.4), (3.14) and (3.18) with coefficients given by (3.15) – (3.17) is a closed stochastic system which can be considered independently of (1.1).

At the next step starting with the system (3.4), (3.14) and (3.19) which can be rewritten in the form

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with coefficients $\tilde{c}^{q},C^{q}$ given by (3.16)– (3.18) and

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we will formulate conditions to ensure that the functions $u^{q}(t,x)$ defined by the last relation exist and give the required generalized solution of the problem (1.2), (1.3). This will be done in a forthcoming paper.

Acknowledgement The financial support of the RNF Grant 17-11-01136 is gratefully acknowledged.