Introduction and Main Results

Abstract

This book is devoted to the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes. The purpose of the book provides a careful and accessible exposition of the functional analytic approach to the problem of construction of Markov processes with boundary conditions in probability theory. We construct a Feller semigroup 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. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. The following diagram gives a bird’s eye view of Markov processes, semigroups and boundary value problems and how these relate to each other: