Boundary value problems and Markov processes /

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Bibliographic Details
Author / Creator:Taira, Kazuaki.
Edition:2nd ed.
Imprint:Berlin : Springer, ©2009.
Description:1 online resource (xii, 186 pages) : illustrations.
Series:Lecture notes in mathematics, 0075-8434 ; 1499
Lecture notes in mathematics (Springer-Verlag) ; 1499.
Subject:Boundary value problems.
Differential equations, Elliptic.
Markov processes.
Boundary value problems.
Differential equations, Elliptic.
Markov processes.
Problèmes aux limites.
Équations différentielles elliptiques.
Processus de Markov.
Semilineare parabolische Differentialgleichung.
Elliptischer Differentialoperator.
Electronic books.
Format: E-Resource Book
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Notes:"This second edition has been revised to streamline some of the analysis and to give better coverage of important examples and applications. The errors in the first printing are corrected ... additional references have been included in the bibliography"--Page vii.
Includes bibliographical references (pages 179-182) and index.
Print version record.
Summary:Annotation 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.
Other form:Print version: Taira, Kazuaki. Boundary value problems and Markov processes. Berlin : Springer, ©2009 9783642016769