Temporal regularity for the stochastic heat equation with rough dependence in space

TL;DR

This paper investigates the temporal regularity and asymptotic behavior of solutions to a stochastic heat equation driven by rough spatial noise, deriving laws of iterated logarithm and variation properties.

Contribution

It provides new insights into the temporal regularity and asymptotic properties of solutions with rough spatial dependence, extending previous well-posedness results.

Findings

Established Khintchine's law of iterated logarithm for the solution

Proved Chung's law of iterated logarithm for the temporal process

Analyzed the $q$-variations of the solution over time

Abstract

Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ \sigma(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac 14,\frac 12)$ in the space variable. When $\sigma(0)=0$, the well-posedness of the solution and its H\"older continuity have been proved by Hu et al. \cite{HHLNT2017}. In this paper, we study the asymptotic properties of the temporal gradient $u(t+\varepsilon, x)-u(t, x)$ at any fixed $t \ge 0$ and $x\in \mathbb R$, as $\varepsilon\downarrow 0$. As applications, we deduce Khintchine's law of iterated logarithm, Chung's law of iterated logarithm, and a result on the $q$-variations of the temporal process $\{u(t, x)\}_{t \ge 0}$, where $x\in \mathbb R$ is fixed.