Boundary Value Problems and Markov Processes (Lecture Notes in Mathematics (1499)) - Softcover

Taira, Kazuaki

Softcover

ISBN 10: 3642016766 ISBN 13: 9783642016769

Publisher: Springer, 2009

Synopsis

Thismonographisanexpandedandrevisedversionofasetoflecturenotesfor thegraduatecoursesgivenbytheauthorbothatHiroshimaUniversity(1995- 1997) and at the University of Tsukuba (1998-2000)which were addressed to the advanced undergraduates and beginning-graduate students with interest in functional analysis, partial di?erential equations and probability. The ?rst edition of this monograph, which was based on the lecture notes given at the University of Tsukuba (1988-1990), was published in 1991. This edition was found useful by a number of people, but it went out of print after afewyears. Thissecondeditionhasbeenrevisedtostreamlinesomeoftheanalysisand to give better coverage of important examples and applications. The errors in the ?rst printing are corrected thanks to kind remarks of many friends. In order to make the monograph more up-to-date, additional references have been included in the bibliography. This secondedition may be consideredas a short introduction to the more advanced book "Semigroups, boundary value problems and Markov processes" which was published in the Springer Monographs in Mathematics series in 2004.For graduatestudents workingin functionalanalysis, partialdi?erential equations and probability, it may serve as an e?ective introduction to these three interrelated ?elds of analysis. For graduate students about to major in the subject and mathematicians in the ?eld looking for a coherent overview, it will provide a method for the analysis of elliptic boundary value problems p in the framework of L spaces. My special thanks go to the editorial sta's of Springer-Verlag for their unfailing helpfulness and cooperation during the production of this second edition. This research was partially supported by Grant-in-Aid for General Sci- ti?cResearch(No.19540162), MinistryofEducation, Culture, Sports, Science and Technology, Japan.

About this title

From the Back Cover

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.

Edition details

• Publisher: Springer
• Publication date: 2009
• Language: English
• ISBN 10: 3642016766
• ISBN 13: 9783642016769
• Binding: Paperback
• Edition number: 2
• Number of pages: 204