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Hitting Probabilities for Nonlinear Systems of Stochastic Waves
 
Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland and University of Barcelona, Barcelona, Spain
Marta Sanz-Solé University of Barcelona, Barcelona, Spain
Hitting Probabilities for Nonlinear Systems of Stochastic Waves
eBook ISBN:  978-1-4704-2507-4
Product Code:  MEMO/237/1120.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
Hitting Probabilities for Nonlinear Systems of Stochastic Waves
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Hitting Probabilities for Nonlinear Systems of Stochastic Waves
Robert C. Dalang Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland and University of Barcelona, Barcelona, Spain
Marta Sanz-Solé University of Barcelona, Barcelona, Spain
eBook ISBN:  978-1-4704-2507-4
Product Code:  MEMO/237/1120.E
List Price: $70.00
MAA Member Price: $63.00
AMS Member Price: $42.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2372015; 75 pp
    MSC: Primary 60;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Upper bounds on hitting probabilities
    • 3. Conditions on Malliavin matrix eigenvalues for lower bounds
    • 4. Study of Malliavin matrix eigenvalues and lower bounds
    • A. Technical estimates
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
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Volume: 2372015; 75 pp
MSC: Primary 60;

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.

  • Chapters
  • 1. Introduction
  • 2. Upper bounds on hitting probabilities
  • 3. Conditions on Malliavin matrix eigenvalues for lower bounds
  • 4. Study of Malliavin matrix eigenvalues and lower bounds
  • A. Technical estimates
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
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