Hitting probabilities for non-linear systems of stochastic waves

TL;DR

This paper investigates hitting probabilities for solutions to non-linear stochastic wave systems in low dimensions, establishing bounds using Malliavin calculus and exploring conditions for points to be polar or non-polar.

Contribution

It provides new bounds on hitting probabilities for non-linear stochastic wave equations driven by Gaussian noise, linking geometric properties of sets to the solution's behavior.

Findings

Points are polar when $d(2-eta) > 2(k+1)$.

Points are not polar in low dimensions.

Open interval exists where polarity of points remains unresolved.

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 $k \in \{1,2,3\}$, 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 $\beta$. 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 $d(2-\beta) > 2(k+1)$, 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.