On a class of stochastic partial differential equations

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• 作者：Jian Song ^{txjsong@hku.hk}
• 关键词：Stochastic partial differential equation ; Gaussian noise ; Feynman&ndash ; Kac formula ; Malliavin calculus
• 刊名：Stochastic Processes and their Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：127
• 期：1
• 页码：37-79
• 全文大小：488 K
• 卷排序：127

文摘

This paper concerns the stochastic partial differential equation with multiplicative noise ∂u∂t=Lu+uẆ, where LL is the generator of a symmetric Lévy process XX, 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)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.