On a class of stochastic partial differential equations

This paper concerns the stochastic partial differential equation with multiplicative noise ∂u∂t=Lu+uẆ, where L is the generator of a symmetric Lévy process X, Ẇ is a Gaussian noise and uẆ is understood both in the senses of Stratonovich and Skorohod. The Feynman-Kac type of representations for the solutions and the moments of the solutions are obtained, and the Hölder continuity of the solutions is also studied. As a byproduct, when γ(x) is a nonnegative and nonnegative-definite function, a sufficient and necessary condition for ∫0t∫0t|r−s|−β0γ(Xr−Xs)drds to be exponentially integrable is obtained.