Local times for systems of non-linear stochastic heat equations

We consider u(t, x) = (u1(t, x) , ⋯ , ud(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≤ 3 , the local time L(ξ, t) of {u(t,x),t∈[0,T]} exists and L(· , t) belongs a.s. to the Sobolev space Hα(Rd) for α<4-d2, and when d≥ 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∈[0,T]}. These results are then used to investigate the irregularity of the coordinate functions of {u(t,x),t∈[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(t1, x) - u(t, x) , ⋯ , u(tn, x) - u(tn-1, x)) , which is a new result and of independent interest.