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

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 Lévy process $X$ taking value in ${\mathbb{R}}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in \left(\frac{1}{2},1\right)$ 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).

References in corpus (2)

• On Fourier transforms of radial functions and distributions
• Feynman--Kac formula for the heat equation driven by fractional noise with Hurst parameter $H<1/2$