Feynman-Kac formula for fractional heat equation driven by fractional white noise

Abstract

In this paper we obtain a Feynman-Kac formula for the solution of a fractional stochastic heat equation driven by fractional noise. One of the main difficulties is to show the exponential integrability of some singular nonlinear functionals of symmetric stable L\'evy motion. This difficulty will be overcome by a technique developed in the framework of large deviation. This Feynman-Kac formula is applied to obtain the H\"older continuity and moment formula of the solution.