Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

TL;DR

This paper studies the stochastic heat equation with multiplicative fractional-colored noise, establishing conditions for the existence of solutions based on the spatial covariance kernel and providing a moment representation involving Brownian motion intersection local times.

Contribution

It introduces new existence criteria for solutions depending on the noise's spatial covariance and fractional properties, and offers a novel moment representation involving intersection local times.

Findings

Solutions exist under specific conditions on the spatial covariance kernel and dimension.

Explicit moment formulas are derived using intersection local times of Brownian motions.

The type of spatial kernel significantly affects the existence and behavior of solutions.

Abstract

We consider the stochastic heat equation with multiplicative noise $u_t={1/2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order $\alpha<d$, then the sufficient condition for the existence of the solution is $d \leq 2+\alpha$ (if $H>1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of $k$ independent $d$-dimensional Brownian motions.