Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises

TL;DR

This paper establishes the existence and uniqueness of mild solutions for a class of stochastic heat equations driven by both white and fractional noises, expanding understanding of such complex stochastic PDEs.

Contribution

It provides the first proof of existence and uniqueness for non-autonomous stochastic heat equations with mixed white and fractional noises under specified conditions.

Findings

Proved existence and uniqueness of mild solutions.

Handled equations with fractional Brownian motion.

Extended results to non-autonomous parabolic SPDEs.

Abstract

We prove the existence and uniqueness of a mild solution for a class of non-autonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb{R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous and the coefficient at the fractional Brownian motion is an affine function.

Equations

$$du(x,t)$$

$$\displaystyle\quad+h\bigl{(}u(x,t)\bigr{)}W^{H}(x,dt),\qquad(x,t)\in D\times(0,T],$$

$$u(x,0)$$

$$\frac{\partial u(x,t)}{\partial n(k)}$$

$$\operatorname{div}\bigl{(}k(x,t)\nabla u(x,t)\bigr{)}=\sum_{i,j=1}^{d}\frac{\partial}{\partial x_{i}}\left(k_{i,j}(x,t)\frac{\partial}{\partial x_{j}}u(x,t)\right);$$

$$\frac{\partial u(x,t)}{\partial n(k)}=\sum _{i,j=1}^d{k}_{i,j}(x,t)n_i(x)\frac{\partial }{\partial x_j}u(x,t),$$

$$u(\cdot ,t)$$

$$+g(u(\cdot ,s))dB(\cdot ,s)+h(u(\cdot ,s))d{B}^H(\cdot ,s),$$

$$\sum _{i,j=1}^d{k}_{i,j}(x,t)q_i{q}_j\ge \underline{k}{\left|q\right|}^2,$$

$${\Vert u\Vert }_{\alpha ,2,T}^2:={\left(\underset{t\in [0,T]}{\sup }{\Vert u(t)\Vert }_2\right)}^2+{\int }_0^T{\left({\int }_0^t\frac{{\Vert u(t)-u(s)\Vert }_2}{(t-s{)}^{\alpha +1}}ds\right)}^{2}dt<\infty .$$

$${\Vert u\Vert }_{\alpha ,1,T}:={\int }_0^T\left(\frac{{\Vert u(t)\Vert }_2}{t^{\alpha }}+{\int }_0^t\frac{{\Vert u(t)-u(s)\Vert }_2}{(t-s{)}^{\alpha +1}}ds\right)dt.$$

$${\Vert f\Vert }_{\alpha ,0;t}=\underset{0\le u<v<t}{\sup }\left(\frac{\left|f(v)-f(u)\right|}{(v-u{)}^{1-\alpha }}+{\int }_u^v\frac{\left|f(u)-f(z)\right|}{(z-u{)}^{2-\alpha }}dz\right).$$

$${\partial }_{t}G(x,t;y,s)$$

$$\frac{\partial G(x,t;y,s)}{\partial n(k)}$$

$${\int }_{D}G(\cdot ,s;y,s)\phi (y)dy$$

$$\left|{\partial }_x^{\gamma }{\partial }_t^{\delta }G(x,t;y,s)\right|\le C(t-s{)}^{-\frac{1}{2}(d+|\gamma |+2\delta )}\exp \left\{-C\frac{{\left|x-y\right|}^2}{t-s}\right\}$$

$$\left|G(x,t;y,s)\right|\le C(t-s{)}^{-\frac{d}{2}}\exp \left\{-C\frac{{\left|x-y\right|}^2}{t-s}\right\}.$$

$$U(t,s)\psi (x)={\int }_{D}G(x,t;y,s)\psi (y)dy.$$

$$|U(t,s)\psi (x)|\le C\mathsf{E}\left|\psi (x+c\sqrt{t-s}\zeta )\text{mathbbm}{1}_D(x+c\sqrt{t-s}\zeta )\right|$$

$$\displaystyle\left\lVert U(t,s)\psi\right\rVert^{2}_{2}\leq C\int_{D}\big{(}\mathsf{E}\left\lvert\psi(x+c\sqrt{t-s}\zeta)\right\rvert\mathbbm{1}_{D}(x+c\sqrt{t-s}\zeta)\big{)}^{2}dx$$

$$\le C\mathsf{E}{\int }_D\psi (x+c\sqrt{t-s}\zeta {)}^2\text{mathbbm}{1}_D(x+c\sqrt{t-s}\zeta )dx\le C\mathsf{E}{\Vert \psi \Vert }_2^2.$$

$$\left|G(x,t;y,v)-G(x,t;y,r)\right|\le (t-v{)}^{-\delta }(v-r{)}^{\delta }(t-t^{\ast }{)}^{-d/2}\exp \left\{-C\frac{{\left|x-y\right|}^2}{t-t^{\ast }}\right\}.$$

$$\left|G(x,t;y,v)-G(x,s;y,v)\right|\le (t-s{)}^{\delta }(s-v{)}^{-\delta }(v^{\ast }-v{)}^{-d/2}\exp \left\{-C\frac{{\left|x-y\right|}^2}{v^{\ast }-v}\right\}.$$

$$\left|G(x,t;y,v)-G(x,s;y,v)-G(x,t;y,r)-G(x,s;y,r)\right|$$

$$\le (t-s{)}^{\delta }(s-v{)}^{-1}(v-r{)}^{1-\delta }$$

$$\times \left((v^{\ast }-v{)}^{-d/2}\exp \left\{-C\frac{{\left|x-y\right|}^2}{v^{\ast }-v}\right\}+(r^{\ast }-r{)}^{-d/2}\exp \left\{-C\frac{{\left|x-y\right|}^2}{r^{\ast }-r}\right\}\right).$$

$$W(\cdot ,t)=\sum _{j=1}^{\infty }{\lambda }_j^{1/2}{e}_j(\cdot )B_j(t),$$

$$W^H(\cdot ,t)=\sum _{j=1}^{\infty }{\mu }_j^{1/2}{e}_j(\cdot )B_j^H(t),$$

$${\int }_0^t\Phi (s)\psi (s)dW(s):=\sum _{j=1}^{\infty }{\lambda }_j^{1/2}{\int }_0^t\Phi (s)(\psi (s)e_j)d{B}_j(s),$$

$$\mathsf{E}{\Vert {\int }_0^t\Phi (s)\psi (s)dW(s)\Vert }_2^2=\mathsf{E}\sum _{j=1}^{\infty }{\lambda }_j{\int }_0^t{\Vert \Phi (s)(\psi (s)e_j)\Vert }_2^{2}ds,$$

$$\mathsf{E}\underset{0\le s\le t}{\sup }{\Vert {\int }_0^s\Phi (v)\psi (v)dW(v)\Vert }_2^p\le C_p\mathsf{E}{\left[\sum _{j=1}^{\infty }{\lambda }_j{\int }_0^t{\Vert \Phi (s)(\psi (s)e_j)\Vert }_2^{2}ds\right]}^{p/2}.$$

$$S(\cdot ,t)={\int }_0^{t}U(t,s)\psi (s)dW(s)=\sum _{j=1}^{\infty }{\lambda }_j^{1/2}{\int }_0^t{\int }_{D}G(\cdot ,t,y,s)\psi (y,s)e_j(y)dyd{B}_j(s).$$

$$\mathsf{E}\underset{0\le s\le t}{\sup }{\Vert S(\cdot ,s)\Vert }_2^p\le C_p^{\prime }\mathsf{E}{\left[{\int }_0^t{\Vert \psi (\cdot ,s)\Vert }_2^{2}ds\right]}^{p/2}.$$

$${\mathcal{D}}_{a+}^{\alpha }f(x)$$

$${\mathcal{D}}_{b-}^{1-\alpha }g(x)$$

Full text

1. Introduction

Let $(\Omega,\mathcal{F},\mathop{\kern 0.0pt\mathsf{P}}\nolimits)$ be a complete probability space. For a fixed $T>0$ let $\mathbb{F}=\left\{\mathcal{F}\right\}_{t\in[0,T]}$ be a filtration satisfying the standard assumptions. Let $\beta\in(0,1)$ be fixed throughout the paper. Assume that $D\subset\mathbb{R}^{d}$ is a bounded domain with boundary $\partial D$ of class $C^{2+\beta}$.

We consider the following stochastic partial differential equation with boundary conditions

[TABLE]

Here $W$ is an $L^{2}(D)$-valued Wiener process and $W^{H}$ is an $L^{2}(D)$-valued fractional Brownian motion with the Hurst index $H\in(1/2,1)$. Furthermore, $k=\left\{k_{i,j}\right\}\colon D\to\mathbb{R}^{d\times d}$ is a matrix-valued field, consequently,

[TABLE]

$n(k)(x)\coloneqq k(x,t)n(x)$ denotes the conormal vector-field, and the relation (3) stands for conormal derivative of $u$ relative to $k$, that is

[TABLE]

where $n(x)\in\mathbb{R}^{d}$ is an outer normal vector to $\partial D$. We are interested in the existence and uniqueness of a mild solution to (1)–(3). The precise statement of the problem and the definition of a mild solution will be given in Sections 2 and 3.

In the pure Wiener case, where $h=0$, the problem was investigated by Sanz-Solé and Vuillermot [11], who introduced three different notions for solutions (namely, variation solutions of the first and the second kind and mild solutions) and showed their indistinguishability. Also, they proved the existence, the uniqueness and the pointwise boundedness of the moments along with the spatial Sobolev regularity of such solutions. Later their results for mild solutions were improved in several directions by Veraar [15, Sec. 8]. In particular, he proved the existence and uniqueness of a mild solution in the case of random coefficients, depending also on $(x,t)$.

The pure “fractional” case, where $g=0$, was studied in [10] and [12]. In [10] the existence, uniqueness and indistinguishability of two types of variational solutions were proved, assuming that the coefficients $f$ and $h$ are Lipschitz continuous and the derivative of $h$ is Hölder continuous. In [12] the authors proved the existence of a mild solution and established its relation with the variational solution of type II from [10] and the Hölder continuity of its sample paths. When $h$ is an affine function, they also proved the uniqueness of a mild solution and the indistinguishability of mild and variational solutions.

In this paper we will consider mild formulation of the stochastic heat equation (1) with white and fractional noises:

[TABLE]

where $U(t,s)=\exp\left\{\int_{s}^{t}A_{u}du\right\}$ is the evolution family on $L^{2}(D)$ corresponding to the elliptic operator A_{t}\psi(\cdot)=\operatorname{div}\big{(}k(\cdot,t)\nabla\psi(\cdot)\big{)} with Neumann boundary conditions (3); the precise formulations will be given later. We show that equation (1) has a unique mild solution. The conditions on the coefficients are similar to those of [11] and [12]. In particular, the functions $f$ and $g$ are assumed to be Lipschitz continuous, and $h$ is assumed to be an affine function. In order to analyze the equation with two different noises (and, consequently, with two different types of stochastic integrals), we replace the fractional Brownian motion by a smooth process, transforming the equation (1) into a stochastic partial differential equation with random drift driven by a Wiener process. This approach was developed for ordinary stochastic differential equations involving both Wiener process and fractional Brownian motion in the article [8]. In [13] it was applied to mixed stochastic delay equations.

We organize this article in the following way. In Section 2, we formulate the assumptions, define $L^{2}(D)$-valued Wiener and fractional Brownian processes and introduce the corresponding stochastic integrals. Also, this section contains some properties of Green’s function associated with our equation. In Section 3 we define a mild solution and prove its existence and uniqueness. In Appendix, we collect auxiliary estimates for solutions that are used for the proving of main result.

2. Preliminaries

2.1. Assumptions on the coefficients and on the initial value

1. (A1)

Assumptions on $k$ and $n$:

1. (i)

$k_{i,j}=k_{j,i}$ for all $i,j=1,\ldots,d$; 2. (ii)

$k_{i,j}\in C^{\beta,\beta^{\prime}}(\overline{D}\times[0,T])$ for some $\beta^{\prime}\in(\frac{1}{2},1]$ and for all $i,j=1,\ldots,d$; 3. (iii)

$\frac{\partial}{\partial x_{l}}k_{i,j}\in C^{\beta,\beta/2}(\overline{D}\times[0,T])$ for all $i,j,l=1,\ldots,d$; 4. (iv)

there exists $\underline{k}>0$ such that

[TABLE]

for all $x\in\overline{D}$, $t\in[0,T]$, $q\in\mathbb{R}^{d}$ (here $\left\lvert\,\cdot\,\right\rvert$ denotes the Euclidean norm in $\mathbb{R}^{d}$); 5. (v)

$\displaystyle(x,t)\mapsto\sum_{i=1}^{d}k_{i,j}(x,t)n_{i}(x)\in C^{1+\beta,(1+\beta)/2}(\partial D\times[0,T])$ for each $j$; 6. (vi)

the conormal vector-field $(x,t)\mapsto n(k)(x,t)=k(x,t)n(x)$ is outward pointing, nowhere tangent to $\partial D$ for every $t$. 2. (A2)

The initial condition satisfies $\varphi\in C^{2+\beta}(\overline{D})$ and the conormal boundary condition (3) relative to $k$. 3. (A3)

$f,g\colon\mathbb{R}\to\mathbb{R}$ are Lipschitz continuous functions. 4. (A4)

$h\colon\mathbb{R}\to\mathbb{R}$ is an affine function.

2.2. Norms and spaces

Let $\left\lVert\,\cdot\,\right\rVert_{2}$ and $\left\lVert\,\cdot\,\right\rVert_{\infty}$ be the norms in $L^{2}(D)$ and $L^{\infty}(D)$ respectively. For $\alpha\in(0,1)$ denote by $\mathcal{B}^{\alpha,2}\left(0,T;L^{2}(D)\right)$ the Banach space of Lebesgue-measurable mappings $u\colon[0,T]\to L^{2}(D)$ endowed with the norm

[TABLE]

Denote also for $u\colon[0,T]\to L^{2}(D)$

[TABLE]

For $f\colon[0,T]\to\mathbb{R}$ and $\alpha\in(0,1)$ define a seminorm

[TABLE]

2.3. Green’s function

Let $G\colon\left\{(x,t,y,s):0\leq s<t\leq T,x,y\in\overline{D}\right\}\to\mathbb{R}$ be the parabolic Green’s function associated with the principal part of (1). It is known from [3, 4] that under assumptions (A1) and (A2) $G$ is a continuous function, twice continuously differentiable in $x$, once continuously differentiable in $t$. For every $(y,s)\in D\times(0,T]$ it is a classical solution to the linear initial-boundary value problem

[TABLE]

Moreover, $G$ satisfies the heat kernel estimates

[TABLE]

for $\gamma=(\gamma_{1},\dots,\gamma_{d})$, $\gamma_{1},\dots,\gamma_{d},\delta\in\mathbb{N}\cup\left\{0\right\}$, and $\left\lvert\gamma\right\rvert+2\delta\leq 2$ with $\left\lvert\gamma\right\rvert=\sum_{j=1}^{d}\gamma_{j}$. In particular, for $\left\lvert\gamma\right\rvert=\delta=0$, we have

[TABLE]

We shall refer to (5) as the Gaussian property of $G$.

The evolution family corresponding to the operator A_{t}\psi(\cdot)=\operatorname{div}\big{(}k(\cdot,t)\nabla\psi(\cdot)\big{)} with Neumann boundary conditions (3) is defined as

[TABLE]

From the Gaussian estimates (5) it can be easily shown that the family $U(t,s)$ is of contractive type on $L^{2}(D)$. Indeed, denoting by $\zeta$ a standard Gaussian vector in $\mathbb{R}^{d}$, we have from (5) for $\psi\in L^{2}(D)$

[TABLE]

with some $c>0$, whence

[TABLE]

In the proposition below we collect other useful estimates for $G$, see Eqs. (3.4), (3.5) and (3.36) in [12].

Proposition 2.1** ([12]).**

Under assumption (A1), for all $x,y\in D$ and $\delta\in(\frac{d}{d+2},1)$, $G$ satisfies the following inequalities.

1. (i)

For all $0<r<v<t<T$