Semigroups, Boundary Value Problems and Markov Processes

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.

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.

 

Kazuaki TAIRA is Professor of Mathematics at the University of Tsukuba, Japan, where he has taught since 1998. He received his Bachelor of Science (1969) degree from the University of Tokyo, Japan, and his Master of Science (1972) degree from Tokyo Institute of Technology, Japan, where he served as an Assistant between 1972-1978. He holds the Doctor of Science (1976) degree from the University of Tokyo, and the Doctorat d'Etat (1978) degree from Université de Paris-Sud, France, where he received a French Government Scholarship in 1976-1978. Dr. Taira was also a member of the Institute for Advanced Study, U. S. A., in 1980-1981. He was Associate Professor of the University of Tsukuba between 1981-1995, and Professor of Hiroshima University, Japan, between 1995-1998.

His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.