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# memo_has_moved_text();Hitting probabilities for nonlinear systems of stochastic waves

Robert C. Dalang and Marta Sanz-Solé

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 237, Number 1120
ISBNs: 978-1-4704-1423-8 (print); 978-1-4704-2507-4 (online)
DOI: http://dx.doi.org/10.1090/memo/1120
Published electronically: January 22, 2015
Keywords:Hitting probabilities, systems of stochastic wave equations, spatially homogeneous Gaussian noise, capacity

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Chapters

• Chapter 1. Introduction
• Chapter 2. Upper bounds on hitting probabilities
• Chapter 3. Conditions on Malliavin matrix eigenvalues for lower bounds
• Chapter 4. Study of Malliavin matrix eigenvalues and lower bounds
• Appendix A. Technical estimates

### Abstract

We consider a $d$-dimensional random field $u = \{u(t,x)\}$ that solves a nonlinear 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 $\mathbb {R}^d$, 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.