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

Abstract

In this article we consider the stochastic heat equation $u_t-\Delta u=\dot{B}$ 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 $H>d/4$.

References in corpus (2)

• The Stochastic Heat Equation Driven by a Gaussian Noise: germ Markov Property
• The 1-d stochastic wave equation driven by a fractional Brownian motion

Cited by in corpus (6)

• A Note on a Fenyman-Kac-Type Formula
• $L_p$-Theory for the Stochastic Heat Equation with Infinite-Dimensional Fractional Noise
• Stochastic Heat Equation with Multiplicative Fractional-Colored Noise
• Some linear SPDEs driven by a fractional noise with Hurst index greater than 1/2
• Hitting times for the stochastic wave equation with fractional-colored noise
• On Besov regularity and local time of the stochastic heat equation