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

Abstract

We consider the parabolic Anderson model driven by fractional noise: $$\frac{\partial }{\partial t}u(t,x)=\kappa \mathbf{\Delta }u(t,x)+u(t,x)\frac{\partial }{\partial t}W(t,x)x\in {\mathbb{Z}}^d,t\ge 0,$$ where $\kappa >0$ is a diffusion constant, $\mathbf{\Delta }$ is the discrete Laplacian defined by $\mathbf{\Delta }f(x)=\frac{1}{2d}{\sum }_{|y-x|=1}(f(y)-f(x))$, and $\{W(t,x);t\ge 0{\}}_{x\in {\mathbb{Z}}^d}$ 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 $y\in {\mathbb{Z}}^d$, $\{X(t);t\ge 0\}$ is a simple random walk with jump rate $\kappa$, started at $x\in {\mathbb{Z}}^d$ and independent of the family $\{W(t,x);t\ge 0{\}}_{x\in {\mathbb{Z}}^d}$ and ${\mathbb{E}}^x$ is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter $H\in (0,1)$.