Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

2007/02/23 by Dalang, Robert C., Khoshnevisan, Davar, Nualart, Eulalia · 5 citations
#60J45 #FOS: Mathematics #Primary: 60H15 #Probability (math.PR) #Secondary: 60G60

paper · doi:10.48550/arxiv.math/0702710

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 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.

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