Systems of stochastic Poisson equations: Hitting probabilities

Marta Sanz-Solé and Noèlia Viles

Stochastic Processes and their Applications, 2018, vol. 128, issue 6, 1857-1888

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⊂Rk, 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 of the canonical distance of the process or, equivalently, on L2 estimates of increments of the Green function of the Laplace equation.

Keywords: Systems of stochastic Poisson equations; Hitting probabilities; Capacity; Hausdorff measure (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2017.08.014

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