# Hitting Probabilities for Nonlinear Systems of Stochastic Waves

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*Robert C. Dalang; Marta Sanz-Solé*

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.

#### Table of Contents

# Table of Contents

## Hitting Probabilities for Nonlinear Systems of Stochastic Waves

- Cover Cover11 free
- Title page i2 free
- Chapter 1. Introduction 18 free
- Chapter 2. Upper bounds on hitting probabilities 916 free
- Chapter 3. Conditions on Malliavin matrix eigenvalues for lower bounds 2128
- Chapter 4. Study of Malliavin matrix eigenvalues and lower bounds 3138
- Appendix A. Technical estimates 4754
- Bibliography 7380
- Back Cover Back Cover188