Boundary Value Problems and Markov Processes: Functional Analysis Methods for Markov Processes

ISBN-13: 9783030487874 / Angielski / Miękka / 2020 / 502 str.

Kazuaki Taira

Opis

This is 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. It presents new developments in the theory of singular integrals.

Recenzje

"By reading this book, a broad spectrum of readers will be able to understand and appreciate the mathematical crossroads of functional analysis, boundary value problems, and probability theory as developed in the more advanced books ... . this book provides a compendium for a large variety of facts from functional analysis, pseudo-differential operators, and Markov processes. Indeed, it gives detailed coverage of important examples and applications in this area." (J. A. van Casteren, Mathematical Reviews, October, 2022)

Spis treści

- Preface to the Third Edition. - Preface to the Second Edition. - Introduction and Main Results. - Part I Analytic and Feller Semigroups and Markov Processes. - Analytic Semigroups. - Markov Processes and Feller Semigroups. - Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems. - Lp Theory of Pseudo-Differential Operators. - Boutet de Monvel Calculus. - Lp Theory of Elliptic Boundary Value Problems. - Part III Analytic Semigroups in Lp Sobolev Spaces. - Proof of Theorem 1.2. - A Priori Estimates. - Proof of Theorem 1.4. - Part IV Waldenfels Operators, Boundary Operators and Maximum Principles. - Elliptic Waldenfels Operators and Maximum Principles. - Boundary Operators and Boundary Maximum Principles. - Part V Feller Semigroups for Elliptic Waldenfels Operators. - Proof of Theorem 1.5 - Part (i). - Proofs of Theorem 1.5, Part (ii) and Theorem 1.6. - Proofs of Theorems 1.8, 1.9, 1.10 and 1.11. - Path Functions of Markov Processes via Semigroup Theory. - Part VI Concluding Remarks. - The State-of-the-Art of Generation Theorems for Feller Semigroups. 

O autorze

Kazuaki Taira was a Professor of mathematics at the University of Tsukuba, Japan. He received his Bachelor of Science degree in 1969 from the University of Tokyo and his Master of Science degree in 1972 from the Tokyo Institute of Technology, where he served as an assistant from 1972 to 1978. In 1976 he was awarded the Doctor of Science degree by the University of Tokyo, and in 1978 the Doctorat d'Etat degree by Université de Paris-Sud (Orsay), where he had studied on a French government scholarship (1976–1978).

Taira was also a member of the Institute for Advanced Study (Princeton) (1980–1981), associate professor at the University of Tsukuba (1981–1995), and professor at Hiroshima University (1995–1998). In 1998, he returned to the University of Tsukuba to teach there again as a professor. From 2009 to 2017 he was a part-time professor at Waseda University (Tokyo). His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.

Blurb

This 3rd edition provides an insight into the mathematical crossroads formed by functional analysis (the macroscopic approach), partial differential equations (the mesoscopic approach) and probability (the microscopic approach) via the mathematics needed for the hard parts of Markov processes. It brings these three fields of analysis together, providing a comprehensive study of Markov processes from a broad perspective. The material is carefully and effectively explained, resulting in a surprisingly readable account of the subject.

The main focus is on a powerful method for future research in elliptic boundary value problems and Markov processes via semigroups, the Boutet de Monvel calculus. A broad spectrum of readers will easily appreciate the stochastic intuition that this edition conveys. In fact, the book will provide a solid foundation for both researchers and graduate students in pure and applied mathematics interested in functional analysis, partial differential equations, Markov processes and the theory of pseudo-differential operators, a modern version of the classical potential theory.