Concluding Remarks

Abstract

This book is devoted to a careful and accessible exposition of the functional analytic approach to the problem of construction of Markov processes with Ventcel’ boundary conditions in probability theory. More precisely, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the state space D\M until it “dies” at the time when it reaches the set M where the particle is definitely absorbed (see Figure 11.1). Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of pseudo-differential operators which may be considered as a modern theory of the classical potential theory.