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On a class of stochastic partial differential equations
In this paper, we study the stochastic partial differential equation with
multiplicative noise ,
where is the generator of a symmetric L\'evy process and 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 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 H\"older continuity of the solution. As a
byproduct, when is a nonnegative and nonngetive-definite function,
a sufficient and necessary condition for to be exponentially integrable is
obtained.Comment: 46 page
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