The purpose of this book is to provide a careful and accessible account along modern lines of the subject wh ich the title deals, as weIl as to discuss prob lems of current interest in the field. Unlike many other books on Markov processes, this book focuses on the relationship between Markov processes and elliptic boundary value problems, with emphasis on the study of analytic semigroups. More precisely, this book is devoted to the functional analytic approach to a class of degenerate boundary value problems for second-order elliptic integro-differential operators, called Waldenfels operators, whi:h in cludes as particular cases the Dirichlet and Robin problems. We prove that this class of boundary value problems provides a new example of analytic semi groups both in the LP topology and in the topology of uniform convergence. As an application, we construct a strong Markov process corresponding to such a physical phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it "dies" at the time when it reaches the set where the particle is definitely absorbed. The approach here is distinguished by the extensive use of the techniques characteristic of recent developments in the theory of partial differential equa tions. The main technique used is the calculus of pseudo-differential operators which may be considered as a modern theory of potentials.

New Edition in preparation The book is of great appeal to both advanced students and researchers Serves as an effective introduction to three interrelated subjects of analysis: semigroups, Markov processes and elliptic boundary value problems Provides a new method for the analysis of Markov processes, a powerful method clearly capable of extensive further development Includes supplementary material: sn.pub/extras

Autorentext

Professor Kazuaki TAIRA, born in Tokyo, Japan in 1946, served for many years as a Professor of Mathematics at the University of Tsukuba (1998-2009). He received his Bachelor of Science in 1969 from the University of Tokyo, Japan and his Master of Science in 1972 from Tokyo Institute of Technology, Japan, where he served as an assistant from 1972 to 1978. The Doctor of Science degree was awarded to him in 1976 by the University of Tokyo and in 1978 the Doctorat d'Etat degree was given to him by Universite de Paris-Sud, France. He held a French government scholarship from 1976 to 1978. He has also served as a member of the Institute for Advanced Study, USA (1980-1981), as an Associate Professor at the University of Tsukuba (1981-1995) and as a Professor at Hiroshima University, Japan (1995-1998). In 1998, he accepted an invitation from the University of Tsukuba to teach there again as a Professor. Since 2009, he has also been a part-time Professor at Waseda University. His current research interests are in the study of three interrelated subjects in analysis: semi groups, elliptic boundary value problems and Markov processes.

Inhalt
1 Introduction and Main Results.- 2 Theory of Semigroups.- 3 Markov Processes and Semigroups.- 4 Theory of Distributions.- 5 Theory of Pseudo-Differential Operators.- 6 Elliptic Boundary Value Problems.- 7 Elliptic Boundary Value Problems and Feller Semigroups.- 8 Proof of Theorem 1.1.- 9 Proof of Theorem 1.2.- 10 A Priori Estimates.- 11 Proof of Theorem 1.3.- 12 Proof of Theorem 1.4, Part (i).- 13 Proofs of Theorem 1.5 and Theorem 1.4, Part (ii).- 14 Boundary Value Problems for Waldenfels Operators.- A Boundedness of Pseudo-Differential Operators.- A.1 The Littlewood-Paley Series.- A.2 Definition of Sobolev and Besov Spaces.- A.3 Non-Regular Symbols.- A.5 Proof of Proposition A.7.- A.6 Proof of Proposition A.8.- B Unique Solvability of Pseudo-Differential Operators.- C The Maximum Principle.- C.1 The Weak Maximum Principle.- C.2 The Strong Maximum Principle.- C.3 The Boundary Point Lemma.- References.- Index of Symbols.