Lecture Notes in Mathematics

Boundary Value Problems and Markov Processes

Authors: Taira, Kazuaki

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About this book

This volume is devoted to a thorough and accessible exposition on the functional analytic approach to the problem of construction of Markov processes with Ventcel' boundary conditions in probability theory.  Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called a Waldenfels operator, in the interior of  the domain, and it obeys a boundary condition, called the Ventcel' boundary condition, on the boundary of the domain.  Probabilistically, a Markovian particle moves both by jumps and continuously in the state space and it obeys the Ventcel' boundary condition, which consists of six terms corresponding  to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (or viscosity) phenomenon, the jump phenomenon on the boundary, and the inward jump phenomenon from the boundary. In particular, second-order elliptic differential operators are called diffusion operators and describe analytically strong Markov processes with continuous paths in the state space such as Brownian motion.  We observe that second-order elliptic differential operators with smooth coefficients arise naturally in connection with the problem of construction of Markov processes in probability.  Since second-order elliptic differential operators are pseudo-differential operators, we can make use of the theory of pseudo-differential operators as in the previous book: Semigroups, boundary value problems and Markov processes (Springer-Verlag, 2004).

Our approach here is distinguished by its extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. Several recent developments in the theory of singular integrals have made further progress in the study of elliptic boundary value problems and hence in the study of Markov processes possible.  The presentation of these new results is the main purpose of this book.

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-642-01677-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $69.99
price for USA
  • ISBN 978-3-642-01676-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Boundary Value Problems and Markov Processes
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
1499
Copyright
2009
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-01677-6
DOI
10.1007/978-3-642-01677-6
Softcover ISBN
978-3-642-01676-9
Series ISSN
0075-8434
Edition Number
2
Number of Pages
XII, 192
Number of Illustrations and Tables
41 b/w illustrations
Topics