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

We consider the parabolic Anderson model driven by fractional noise: partial derivative/partial derivative t u(t,x) = k Delta u(t,x) + u(t,x)partial derivative/partial derivative t W(t,x) x is an element of Z(d), t >= 0, where k > 0 is a diffusion constant, Delta is the discrete Laplacian defined by Delta f (x) = 1/2d Sigma vertical bar y-x vertical bar=1 (f(Y) - f (0)), and {W(t,x) ; t > 0} x is an element of Z(d) is a family of independent fractional Brownian motions with Hurst parameter H is an element of (0,1), indexed by 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 u(t, x) = E-infinity [u(0) (X(t)) exp integral(t)(0) W(ds,X(t-s))], (1) is a mild solution to this problem. Here u(0) (y) is the initial value at site y is an element of Z(d), {X(t); t >= 0} is a simple random walk with jump rate f, started at x E Zd and independent of the family {W(t, x) ; t >= 0}(x is an element of Zd) and E-infinity is expectation withrespect to this random walk. We give a unified argument that works for any Hurst parameter H is an element of (0,1). (C) 2015 Elsevier Inc. All rights reserved.