
Hitting Probabilities for Nonlinear Systems of Stochastic Waves
American Mathematical Society (Publisher)
Will be published approx. on 30. September 2015
Book
Paperback/Softback
75 pages
978-1-4704-1423-8 (ISBN)
Description
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.
More details
Series
Language
English
Place of publication
Providence
United States
Target group
College/higher education
Dimensions
Height: 254 mm
Width: 178 mm
Weight
141 gr
ISBN-13
978-1-4704-1423-8 (9781470414238)
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Schweitzer Classification
Persons
Robert C. Dalang, Ecole Polytechnique Federale de Lausanne, Switzerland, and University of Barcelona, Spain.
Marta Sanz-Sole, University of Barcelona, Spain.
Marta Sanz-Sole, University of Barcelona, Spain.
Content
Introduction
Upper bounds on hitting probabilities
Conditions on Malliavin matrix eigenvalues for lower bounds
Study of Malliavin matrix eigenvalues and lower bounds
Appendix A. Technical estimates
Bibliography
Upper bounds on hitting probabilities
Conditions on Malliavin matrix eigenvalues for lower bounds
Study of Malliavin matrix eigenvalues and lower bounds
Appendix A. Technical estimates
Bibliography