Local times for systems of non-linear stochastic heat equations

Abstract

We consider $u(t,x)=(u_1(t,x),\cdots ,u_d(t,x))$ the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a $d$-dimensional space-time white noise. We prove that, when $d\le 3$, the local time $L(\xi ,t)$ of $\{u(t,x),t\in [0,T]\}$ exists and $L(\bullet ,t)$ belongs a.s. to the Sobolev space $H^{\alpha }({\mathbb{R}}^d)$ for $\alpha <\frac{4-d}{2}$, and when $d\ge 4$, the local time does not exist. We also show joint continuity and establish Hölder conditions for the local time of $\{u(t,x),t\in [0,T]\}$. These results are then used to investigate the irregularity of the coordinate functions of $\{u(t,x),t\in [0,T]\}$. Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of $(u(t_1,x)-u(t_0,x),\cdots ,u(t_n,x)-u(t_{n-1},x))$, which is a new result and of independent interest.