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 \(L(\cdot ,t) \) 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.