Systems of stochastic Poisson equations: hitting probabilities

TL;DR

This paper studies the properties and hitting probabilities of solutions to systems of elliptic stochastic equations driven by white noise in low-dimensional spaces, providing bounds based on geometric measures.

Contribution

It establishes bounds on hitting probabilities for Gaussian solutions of stochastic Poisson systems using Hausdorff measure and Bessel-Riesz capacity, with new estimates on Green function increments.

Findings

Upper and lower bounds on hitting probabilities are derived.

Precise estimates on the Green function increments are provided.

Properties of the solutions and their probability laws are characterized.

Abstract

We consider a $d$-dimensional random field $u=(u(x), x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D\subset \mathbb{R}^k$, 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-Sol\'e, 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 $L^2$ estimates of increments of the Green function of the Laplace equation.