Chung's LIL for the linear stochastic fractional heat equation at origin

TL;DR

This paper establishes Chung's law of the iterated logarithm at the origin for solutions to a linear stochastic fractional heat equation driven by Gaussian noise with fractional Brownian covariance.

Contribution

It provides the first Chung's law of the iterated logarithm result for the fractional stochastic heat equation at the origin, considering fractional Laplacian and fractional Brownian noise.

Findings

Chung's law holds at the origin for the solution.

The result characterizes the precise asymptotic behavior of the solution.

The analysis extends classical results to fractional and Gaussian noise settings.

Abstract

Consider the linear stochastic fractional heat equation with vanishing initial condition: $$ \frac{\partial u (t,x)}{\partial t}=-(-\Delta)^{\frac{\alpha}2}u (t,x) + \dot{W}(t,x),\quad t> 0,\, x\in \mathbb R, $$ where $-(-\Delta)^{\frac{\alpha}{2}}$ denotes the fractional Laplacian with power $\alpha\in (1,2]$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-\alpha}2,1\right)$. We establish Chung's law of the iterated logarithm for the solution at $t=0$.