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Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations
E. B. Dynkin, Cornell University, Ithaca, New York
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University Lecture Series
2004; 120 pp; softcover
Volume: 34
ISBN-10: 0-8218-3682-X
ISBN-13: 978-0-8218-3682-8
List Price: US$32
Member Price: US$25.60
Order Code: ULECT/34
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This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that will be of interest to anyone who works on applications of probabilistic methods to mathematical analysis.

The book is suitable for graduate students and research mathematicians interested in probability theory and its applications to differential equations.

Also of interest by this author is Diffusions, Superdiffusions and Partial Differential Equations in the AMS series, Colloquium Publications.

Readership

Graduate students and research mathematicians interested in probability theory and its applications to differential equations.

Reviews

"This book is written by a well-known specialist in the theory of Markov processes and partial differential equation..."

-- Newsletter of the EMS

Table of Contents

  • Introduction
  • Analytic approach
  • Probabilistic approach
  • \(\mathbb{N}\)-measures
  • Moments and absolute continuity properties of superdiffusions
  • Poisson capacities
  • Basic inequality
  • Solutions \(w_\Gamma\) are \(\sigma\)-moderate
  • All solutions are \(\sigma\)-moderate
  • Appendix A: An elementary property of the Brownian motion
  • Appendix B: Relations between Poisson and Bessel capacities
  • References
  • Subject index
  • Notation index
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