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.
Categories: math.PR