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

We consider the parabolic Anderson model driven by fractional noise:∂∂tu(t,x)=κΔu(t,x)+u(t,x)∂∂tW(t,x)x∈Zd,t≥0, where κ>0κ>0 is a diffusion constant, Δ is the discrete Laplacian defined by Δf(x)=12d∑|y−x|=1(f(y)−f(x)), and {W(t,x);t≥0}x∈Zd is a family of independent fractional Brownian motions with Hurst parameter H∈(0,1)H∈(0,1), indexed by ZdZd. 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 representationequation(1)u(t,x)=Ex[uo(X(t))exp⁡∫0tW(ds,X(t−s))], is a mild solution to this problem. Here uo(y)uo(y) is the initial value at site y∈Zdy∈Zd, {X(t);t≥0} is a simple random walk with jump rate κ  , started at x∈Zdx∈Zd and independent of the family {W(t,x);t≥0}x∈Zd and ExEx is expectation with respect to this random walk. We give a unified argument that works for any Hurst parameter H∈(0,1)H∈(0,1).