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

Abstract

We consider a system of d coupled non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution {u(t, x)}t∈R+, x∈[0 ,1], in terms of respectively Hausdorff measure and Newtonian capacity. We determine the Hausdorff dimensions of level sets and their projections. We also present an anisotropic form of the Kolmogorov continuity theorem.