Local times for systems of non-linear stochastic heat equations

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 $d\leq 3$, 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 $d\geq 4$, the local time does not exist. We also show joint continuity and establish H\"{o}lder 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.