, Volume 143, Issue 1-2, pp 285-328
Date: 04 Jan 2008

Stochastic heat equation driven by fractional noise and local time

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Abstract

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 ∈ (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.

Y. Hu is supported by the National Science Foundation under DMS0504783.
D. Nualart is supported by the National Science Foundation under DMS0604207.