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 $\{u(t,x){\}}_{t\in {\mathbb{R}}_{+},x\in [0,1]}$, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.

44 pages; submitted for publication

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