Hitting Probabilities for Nonlinear Systems of Stochastic Waves (Memoirs of the American Mathematical Society)

By Robert C Dalang

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.

• Format: Paperback
• Language: English
• ISBN: 1470414236
• ISBN13: 9781470414238
• Release Date: September 2015
• Publisher: Amer Mathematical Society
• Length: 75 Pages
• Weight: 0.31 lbs.
• Dimensions: 7.0" x 10.0"