Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise

Abstract

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t, x), and upper bounds of Gaussian-type on the two-point density of (u(s, y),u(t, x)). In particular, this estimate quantifies how this density degenerates as (s, y) → (t, x). From these results, we deduce upper and lower bounds on hitting probabilities of the process $${\{u(t,x)\}_{t \in \mathbb{R}_+, x\in [0,1]}}$$ , in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d ≥ 7 and are not polar when d ≤ 5. We also show that the Hausdorff dimension of the range of the process is 6 when d > 6, and give analogous results for the processes $${t \mapsto u(t,x)}$$ and $${x \mapsto u(t,x)}$$ . Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.