On a class of stochastic partial differential equations

Jian Song

Published 2015-03-23Version 1

In this paper, we study the stochastic partial differential equation with multiplicative noise $\frac{\partial u}{\partial t} =\mathcal L u+u\dot W$, where $\mathcal L$ is the generator of a symmetric L\'evy process $X$ and $\dot W$ is a Gaussian noise. For the equation in the Stratonovich sense, we show that the solution given by a Feynman-Kac type of representation is a mild solution, and we establish its H\"older continuity and the Feynman-Kac formula for the moments of the solution. For the equation in the Skorohod sense, we obtain the condition for the existence and uniqueness of the mild solution under which we get Feymnan-Kac formula for the moments of the solution, and we also investigate the H\"older continuity of the solution. As a byproduct, when $\gamma(x)$ is a nonnegative and nonngetive-definite function, a sufficient and necessary condition for $\int_0^t\int_0^t |r-s|^{-\beta_0}\gamma(X_r-X_s)drds$ to be exponentially integrable is obtained.