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

Kazuaki Taira

Softcover

ISBN 10: 354054996X ISBN 13: 9783540549963

Publisher: Springer, 1992

Synopsis

Focussing on the interrelations of the subjects of Markov processes, analytic semigroups and elliptic boundary value problems, this monograph provides a careful and accessible exposition of functional methods in stochastic analysis. The author studies a class of boundary value problems for second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems, and proves that this class of boundary value problems provides a new example of analytic semigroups both in the Lp topology and in the topology of uniform convergence. As an application, one can construct analytic semigroups corresponding to the diffusion phenomenon of a Markovian particle moving continuously in the state space until it "dies", at which time it reaches the set where the absorption phenomenon occurs. A class of initial-boundary value problems for semilinear parabolic differential equations is also considered. This monograph will appeal to both advanced students and researchers as an introduction to the three interrelated subjects in analysis, providing powerful methods for continuing research.

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: 1992
• ISBN 10: 354054996X
• ISBN 13: 9783540549963
• Binding: Paperback
• Language: English
• Edition number: 1
• Number of pages: 144