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Stochastic heat equation driven by fractional noise and local time. (English) Zbl 1152.60331
Summary: The aim of this paper is to study the \(d\)-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter \(H \in (0,1)\) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
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