Real Analysis Methods for Markov Processes: Singular Integrals and Feller Semigroups

by Kazuaki Taira

Overview

This book is devoted to real analysis methods for the problem of constructing Markov processes with boundary conditions in probability theory. Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called the Waldenfels operator, in the interior of the domain, and it obeys a boundary condition, called the Ventcel (Wentzell) boundary condition, on the boundary of the domain. Most likely, a Markovian particle moves both by continuous paths and by jumps in the state space and obeys the Ventcel boundary condition, which consists of six terms corresponding to diffusion along the boundary, an absorption phenomenon, a reflection phenomenon, a sticking (or viscosity) phenomenon, and a jump phenomenon on the boundary and an inward jump phenomenon from the boundary. More precisely, we study a class of first-order Ventcel boundary value problems for second-order elliptic Waldenfels integro-differential operators. By using the Calderón–Zygmund theory of singular integrals, we prove the existence and uniqueness of theorems in the framework of the Sobolev and Besov spaces, which extend earlier theorems due to Bony–Courrège–Priouret to the vanishing mean oscillation (VMO) case. Our proof is based on various maximum principles for second-order elliptic differential operators with discontinuous coefficients in the framework of Sobolev spaces.

My approach is distinguished by the extensive use of the ideas and techniques characteristic of recent developments in the theory of singular integral operators due to Calderón and Zygmund. Moreover, we make use of an Lp variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups by me. The present book is amply illustrated; 119 figures and 12 tables are provided in such a fashion that a broad spectrum of readers understand our problem and main results.

Product Details

ISBN-13:	9789819736584
Publisher:	Springer Nature Singapore
Publication date:	09/03/2024
Edition description:	2024
Pages:	749
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Dr. TAIRA, Kazuaki, born in Tokyo, Japan, on January 1, 1946, was a professor of mathematics at the University of Tsukuba, Japan (1998–2009). He received his Bachelor of Science degree in 1969 from the University of Tokyo, Japan, and his Master of Science degree 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 on June 21, 1976, by the University of Tokyo, and on June 13, 1978, the Doctorat d'Etat degree was given to him by Universit\'{e} de Paris-Sud (Orsay), France. He had been studying there on the French government scholarship from 1976 to 1978.

Dr. TAIRA was also a member of the Institute for Advanced Study (Princeton), USA (1980–1981), was an associate professor at the University of Tsukuba (1981–1995), and a professor at Hiroshima University, Japan (1995–1998). In 1998, he accepted the offer from the University of Tsukuba to teach there again as a professor. He was a part-time professor at Waseda University (Tokyo), Japan, from 2009 to 2017.

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

Table of Contents

Introduction and Main Results.- Elements of Functional Analysis.- Elements of Measure Theory and Lp Spaces.- Elements of Real Analysis.- Harmonic Functions and Poisson Integrals.- Besov Spaces via Poisson Integrals.- Sobolev and Besov Spaces.- Maximum Principles in Sobolev Spaces.- Elements of Singular Integrals.- Calder´on–Zygmund Kernels and Their Commutators.- Calder´on–Zygmund Variable Kernels and Their Commutators.- Dirichlet Problems in Sobolev Spaces.- Calder´on–Zygmund Kernels and Interior Estimates.- Calder´on–Zygmund Kernels and Boundary Estimates.- Unique Solvability of the Homogeneous Dirichlet Problem.- Regular Oblique Derivative Problems in Sobolev Spaces.- Oblique Derivative Boundary Conditions.- Boundary Representation Formula for Solutions.- Boundary Regularity of Solutions.- Proof of Theorems 16.1 and 16.2.- Markov Processes and Feller Semigroups.- Feller Semigroups with Dirichlet Condition.- Feller Semigroups with an Oblique Derivative Condition.- Feller Semigroups and Boundary Value Problems.- Feller Semigroups with a First Order Ventcel’ Boundary Condition.- Concluding Remarks.