Feynman–Kac representation for the parabolic Anderson model driven by fractional noise

Abstract

We consider the parabolic Anderson model driven by fractional noise: $$ \frac{\partial}{\partial t}u(t,x)= \kappa \boldsymbol{\Delta} u(t,x)+ u(t,x)\frac{\partial}{\partial t}W(t,x) \qquad x\in\mathbb{Z}^d\;,\; t\geq 0\,, $$ where $\kappa>0$ is a diffusion constant, $\boldsymbol{\Delta}$ is the discrete Laplacian defined by $\boldsymbol{\Delta} f(x)= \frac{1}{2d}\sum_{|y-x|=1}\bigl(f(y)-f(x)\bigr)$, and $\{W(t,x)\;;\;t\geq0\}_{x \in \mathbb{Z}^d}$ is a family of independent fractional Brownian motions with Hurst parameter $H\in(0,1)$, indexed by $\mathbb{Z}^d$. We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation \begin{equation} u(t,x)=\mathbb{E}^x\Bigl[u_o(X(t))\exp \int_0^t W\bigl(\mathrm{d}s, X(t-s)\bigr)\Bigr]\,, \end{equation} is a mild solution to this problem. Here $u_o(y)$ is the initial value at site $y\in\mathbb{Z}^d$, $\{X(t)\;;\;t\geq0\}$ is a simple random walk with jump rate $\kappa$, started at $x \in \mathbb{Z}^d$ and independent of the family $\{W(t,x)\;;\;t\geq0\}_{x\in\mathbb{Z}^d}$ and $\mathbb{E}^x$ is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter $H\in (0,1)$.