On a class of stochastic fractional heat equations

Abstract

For the fractional heat equation ∂ ∂ t u ( t , x ) = − ( − Δ ) α 2 u ( t , x ) + u ( t , x ) W ˙ ( t , x ) \frac {\partial }{\partial t} u(t,x) = -(-\Delta )^{\frac {\alpha }{2}}u(t,x)+ u(t,x)\dot W(t,x) where the covariance function of the Gaussian noise W ˙ \dot W is defined by the heat kernel, we establish Feynman-Kac formulae for both Stratonovich and Skorohod solutions, along with their respective moments. In particular, we prove that d > 2 + α d>2+\alpha is a sufficient and necessary condition for the equation to have a unique square-integrable mild Skorohod solution. One motivation lies in the occurrence of this equation in the study of a random walk in random environment which is generated by a field of independent random walks starting from a Poisson field.