A Note on a Fenyman-Kac-Type Formula

TL;DR

This paper derives a probabilistic representation for the second moment of solutions to a fractional, colored-in-space stochastic heat equation, extending previous formulas to account for fractional temporal noise using a planar Poisson process.

Contribution

It introduces a novel Feynman-Kac-type formula for stochastic heat equations with fractional time noise and spatial coloring, using a planar Poisson process for the representation.

Findings

Provides a probabilistic representation for second moments of solutions.

Extends existing formulas to fractional, colored-in-space noise.

Utilizes planar Poisson process for the representation.

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.