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

Abstract

Consider the nonlinear stochastic heat equation $$\frac{\partial u(t,x)}{\partial t}=\frac{{\partial }^{2}u(t,x)}{\partial x^2}+\ddot{I}ƒ(u(t,x))\dot{W}(t,x),t>0,x\in \mathbb{R},$$ where $\dot{W}$ is a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. When $\ddot{I}ƒ(0)=0$, the well-posedness of the solution and its Hölder continuity have been proved by Hu et al. \cite{HHLNT2017}. In this paper, we study the asymptotic properties of the temporal gradient $u(t+\epsilon ,x)-u(t,x)$ at any fixed $t\ge 0$ and $x\in \mathbb{R}$, as $\epsilon \downarrow 0$. 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.

31 pages. Comments welcome!