Dalang R.C., Sanz-Sole M. Hitting Probabilities for Nonlinear Systems of Stochastic Waves

Providence: American Mathematical Society, 2015. — 88 p.The authors consider a dd-dimensional random field u={u(t,x)}u={u(t,x)} that solves a non-linear system of stochastic wave equations in spatial dimensions k∈{1,2,3}k∈{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 ββ. Using Malliavin calculus, they establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of RdRd, 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−β)>2(k+1)d(2−β)>2(k+1), points are polar for uu. Conversely, in low dimensions dd, points are not polar. There is, however, an interval in which the question of polarity of points remains open.