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

Abstract

In this article we consider the stochastic heat equation $u_{t}-Δu=\dot B$ 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 $α\in (0,d)$ this condition is $H>(d-α)/4$, 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 $H>d/4$.