ON A CLASS OF STOCHASTIC FRACTIONAL HEAT EQUATIONS

For the fractional heat equation partial derivative/partial derivative t u(t, x) = -(-Delta)(alpha/2) u(t, x) + u(t, x) W-center dot(t, x) where the covariance function of the Gaussian noise W-center dot 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.