The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution

TL;DR

This paper establishes the precise conditions on the Hurst parameter for the existence of solutions to a stochastic heat equation driven by fractional-colored Gaussian noise, extending known results for white space noise.

Contribution

It provides necessary and sufficient conditions for solution existence based on the Hurst index and the spatial covariance kernel, generalizing previous white noise results.

Findings

Solution exists if H > (d - α)/4 for Riesz kernel f.

Condition remains H > d/4 for Bessel and heat kernels.

Relaxation of conditions compared to white space noise case.

Abstract

In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$.