Hitting probabilities for nonlinear systems of stochastic waves

The authors consider a d-dimensional random field u = \{u(t,x)\} that solves a non-linear system of stochastic wave equations in spatial dimensions k \in \{1,2,3\}, driven by a spatially homogeneous Gaussian noise that is white in time. They mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent \beta. Using Malliavin calculus, they establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of \mathbb{R}^d, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that ap