Download or read book Hitting Probabilities for Nonlinear Systems of Stochastic Waves written by Robert C. Dalang and published by . This book was released on 2015 with total page 75 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider a d-dimensional random field u={u(t,x)} that solves a nonlinear system of stochastic wave equations in spatial dimensions k∈{1,2,3}, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent β. Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of Rd, 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), points are polar for u. Conversely, in low dimensions d d, points are not polar. There is however an interval in which the question of polarity of points remains open.