On a class of stochastic fractional heat equations

Abstract

For the fractional heat equation $\frac{\partial }{\partial t}u(t,x)=-(-\Delta {)}^{\frac{\alpha }{2}}u(t,x)+u(t,x)\dot{W}(t,x)$ where the covariance function of the Gaussian noise $\dot{W}$ 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 $d<2+\alpha$ 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.