On a class of stochastic partial differential equations

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 Lévy 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 Hölder continuity and the Feynman-Kac formula for the moments of the solution. For the equation in the Skorohod sense, we obtain a sufficient 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 Hölder continuity of the solution. As a byproduct, when $γ(x)$ is a nonnegative and nonngetive-definite function, a sufficient and necessary condition for $\int_0^t\int_0^t |r-s|^{-β_0}γ(X_r-X_s)drds$ to be exponentially integrable is obtained.

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