Systems of stochastic Poisson equations: hitting probabilities

2016/12/14 by Marta Sanz–Solé, Sanz-Solé, Marta, Noèlia Viles +1 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #Stochastic processes and financial applications #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1612.04567

Abstract

We consider a d-dimensional random field u=(u(x), x∈ D) that solves a system of elliptic stochastic equations on a bounded domain D⊂ ℝ^k, with additive white noise and spatial dimension k=1,2,3. Properties of u and its probability law are proved. For Gaussian solutions, using results from [Dalang and Sanz-Solé, 2009], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on L² estimates of increments of the Green function of the Laplace equation.

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• Pathwise regularity of solutions for a class of elliptic SPDEs with symmetric Lévy noise

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