On a class of stochastic partial differential equations

This paper concerns the stochastic partial differential equation with multiplicative noise partial derivative u/partial derivative t = Lu + u(W) over dot, where L is the generator of a symmetric Levy process X, (W) over dot is a Gaussian noise and u(W) over dot 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 Holder continuity of the solutions is also studied. As a byproduct, when gamma(x) is a nonnegative and nonnegative-definite function, a sufficient and necessary condition for integral(t)(0) integral(t)(0) vertical bar r-s vertical bar(-beta 0)gamma (X-r - X-s)drds to be exponentially integrable is obtained. (C) 2016 Elsevier B.V. All rights reserved.