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2009•arXiv (Cornell University)Open access

A Note on a Fenyman-Kac-Type Formula

Raluca M. Balan

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Abstract

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

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In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

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Available abstract

In this article, we establish a probabilistic representation for the second-order moment of the solution of stochastic heat equation in $[0,1] \times \bR^d$, with multiplicative noise, which is fractional in time and colored in space. This representation is similar to the one given in Dalang, Mueller and Tribe (2008) in the case of an s.p.d.e. driven by a Gaussian noise, which is white in time. Unlike the formula of Dalang, Mueller and Tribe (2008), which is based on the usual Poisson process, our representation is based on the planar Poisson process, due to the fractional component of the noise.

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