Hitting Probabilities for Nonlinear Systems of Stochastic Waves Paperback / softback

by Robert C. Dalang, Marta Sanz-Sole

Part of the Memoirs of the American Mathematical Society series

Paperback / softback

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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 $\mathbb{R}^d$, 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.