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Systems of stochastic Poisson equations: Hitting probabilities

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  • Sanz-Solé, Marta
  • Viles, Noèlia

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.

Suggested Citation

  • Sanz-Solé, Marta & Viles, Noèlia, 2018. "Systems of stochastic Poisson equations: Hitting probabilities," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 1857-1888.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:6:p:1857-1888
    DOI: 10.1016/j.spa.2017.08.014
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    Cited by:

    1. Hinojosa-Calleja, Adrián, 2023. "Exact uniform modulus of continuity for q-isotropic Gaussian random fields," Statistics & Probability Letters, Elsevier, vol. 197(C).
    2. Dalang, Robert C. & Pu, Fei, 2021. "Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations," Stochastic Processes and their Applications, Elsevier, vol. 131(C), pages 359-393.

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