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

Robert C. Dalang·Davar Khoshnevisan·Eulalia Nualart
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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.

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Robert C. Dalang, Davar Khoshnevisan, Eulalia Nualart. 2007-02-23. Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise. https://arxiv.org/abs/math/0702710

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