Sharp Space-Time Regularity of the Solution to Stochastic Heat Equation Driven by Fractional-Colored Noise

TL;DR

This paper investigates the regularity properties of solutions to a stochastic heat equation driven by fractional-colored noise, establishing precise continuity moduli and laws of iterated logarithm in both time and space.

Contribution

It provides new regularity results, including exact modulus of continuity and Chung-type laws, for solutions driven by fractional-colored Gaussian noise, extending previous work.

Findings

Established existence of solutions to the stochastic heat equation.

Derived exact uniform modulus of continuity for the solution.

Proved Chung-type law of iterated logarithm for the solution.

Abstract

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a L\'evy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance function $f$ which is the Fourier transform of a tempered measure $\mu.$ After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution $\{u(t,x),\, t\ge 0,\, x\in\mathbb{R}^d\}$ in both time and space variables. Under mild conditions, we give the exact uniform modulus of continuity and a Chung-type law of iterated logarithm for the sample function $(t,x)\mapsto u(t,x)$. Our results generalize and strengthen the corresponding results of Balan and Tudor (2008) and Tudor and Xiao (2017).