Boundary Value Problems and Markov Processes

  • Authors
  • KazuakiĀ Taira
Part of the Lecture Notes in Mathematics book series (LNM, volume 1499)

Table of contents

  1. Front Matter
    Pages 1-9
  2. Kazuaki Taira
    Pages 1-12
  3. Kazuaki Taira
    Pages 13-54
  4. Kazuaki Taira
    Pages 55-75
  5. Kazuaki Taira
    Pages 87-93
  6. Kazuaki Taira
    Pages 95-100
  7. Kazuaki Taira
    Pages 101-111
  8. Kazuaki Taira
    Pages 113-124
  9. Kazuaki Taira
    Pages 125-159
  10. Kazuaki Taira
    Pages 169-174
  11. Back Matter
    Pages 1-17

About this book

Introduction

This volume is devoted to 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.  Analytically, a Markovian particle in a domain of Euclidean space is governed by an integro-differential operator, called a Waldenfels operator, in the interior of  the domain, and it obeys a boundary condition, called the Ventcel' boundary condition, on the boundary of the domain.  Probabilistically, a Markovian particle moves both by jumps and continuously in the state space and it obeys the Ventcel' boundary condition, which consists of six terms corresponding  to the diffusion along the boundary, the absorption phenomenon, the reflection phenomenon, the sticking (or viscosity) phenomenon, the jump phenomenon on the boundary, and the inward jump phenomenon from the boundary. In particular, second-order elliptic differential operators are called diffusion operators and describe analytically strong Markov processes with continuous paths in the state space such as Brownian motion.  We observe that second-order elliptic differential operators with smooth coefficients arise naturally in connection with the problem of construction of Markov processes in probability.  Since second-order elliptic differential operators are pseudo-differential operators, we can make use of the theory of pseudo-differential operators as in the previous book: Semigroups, boundary value problems and Markov processes (Springer-Verlag, 2004).

Our approach here is distinguished by its extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. Several recent developments in the theory of singular integrals have made further progress in the study of elliptic boundary value problems and hence in the study of Markov processes possible.  The presentation of these new results is the main purpose of this book.

Keywords

Boundary value problem Feller semigroup Markov process Probability theory analytic semigroup elliptic boundary value problem semilinear parabolic equation

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-01677-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 2009
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-01676-9
  • Online ISBN 978-3-642-01677-6
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692