Diffusion Processes and Partial Differential Equations

Abstract

This chapter presents a study the intimate connections between second-order differential operators and Markov processes. It is divided into two chapters. The purpose of the first chapter is to reveal the underlying analytical mechanism of propagation of maximum (sharp maximum principle) for degenerate elliptic operators of second order. The mechanism of propagation of maximum is closely related to the diffusion phenomenon of Markovian particles. The second chapter is devoted to the semigroup approach to the problem of construction of Markov processes in probability theory. It is well-known that by virtue of the celebrated Hille–Yosida theorem in the theory of semigroups, the problem of construction of Markov processes can be reduced to the study of boundary value problems for degenerate elliptic operators of second order. Several recent developments in the theory of partial differential equations have made possible further progress in the study of boundary value problems and hence of the problem of construction of Markov processes.