On a class of stochastic fractional heat equations

For the fractional heat equation ∂ ∂ t u (t, x) = − (− Δ) α 2 u (t, x) + u (t, x) W ˙ (t, x) t u (t, x) = - (-) ^ { 2}u (t, x) + u (t, x) W (t, x) where the covariance function of the Gaussian noise W ˙ 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+ 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.