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

Dalang, Robert C.; Sanz-Sole, Marta

Softcover

ISBN 10: 1470414236 ISBN 13: 9781470414238

Publisher: American Mathematical Society, 2015

Synopsis

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.

About this title

About the Author

Robert C. Dalang, Ecole Polytechnique Federale de Lausanne, Switzerland, and University of Barcelona, Spain.

Marta Sanz-Sole, University of Barcelona, Spain.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2015
• Language: English
• ISBN 10: 1470414236
• ISBN 13: 9781470414238
• Binding: Paperback
• Number of pages: 75