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 $mathbbRd$, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when $d(2-beta) 2(k+1)$, points are polar for $u$. Conversely, in low dimensions $d$, points are not polar. There is, however, an interval in which the question of polarity of points remains open.