Hitting probabilities for non-linear systems of stochastic waves
Abstract
We consider a
d
-dimensional random field
u={u(t,x)}
that solves a non-linear 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 $\IR^d$, 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
, points are not polar. There is however an interval in which the question of polarity of points remains open.