Systems of stochastic Poisson equations: Hitting probabilities

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.