Boundary Value Problems and Markov Processes
by Kazuaki Taira (auth.)

Persian Title: مسائل مقدار مرزی و هیجدهم

Summary and Info
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 integrodifferential 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, secondorder 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 secondorder elliptic differential operators with smooth coefficients arise naturally in connection with the problem of construction of Markov processes in probability. Since secondorder elliptic differential operators are pseudodifferential operators, we can make use of the theory of pseudodifferential operators as in the previous book: Semigroups, boundary value problems and Markov processes (SpringerVerlag, 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.
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