Degenerate diffusion operators arising in population biology / Charles L. Epstein and Rafe Mazzeo.

Format:

Book

Author/Creator:

Epstein, Charles L., 1957-

Contributor:

Mazzeo, Rafe.

Series:

Annals of Mathematics Studies

Annals of mathematics studies ; number 185

Language:

English

Subjects (All):

Elliptic operators.

Markov processes.

Population biology--Mathematical models.

Population biology.

Physical Description:

1 online resource (321 p.)

Edition:

Course Book

Place of Publication:

Princeton : Princeton University Press, 2013.

Language Note:

English

System Details:

Mode of access: World Wide Web.

Summary:

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the Martingale problem and therefore the existence of the associated Markov process. Charles Epstein and Rafe Mazzeo use an "integral kernel method" to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high co-dimensional strata of the boundary. Epstein and Mazzeo establish the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. They show that the semigroups defined by these operators have holomorphic extensions to the right half-plane. Epstein and Mazzeo also demonstrate precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

Contents:

Front matter

Contents

Preface

Chapter 1. Introduction

Part I. Wright-Fisher Geometry and the Maximum Principle

Chapter 2. Wright-Fisher Geometry

Chapter 3. Maximum Principles and Uniqueness Theorems

Part II. Analysis of Model Problems

Chapter 4. The Model Solution Operators

Chapter 5. Degenerate Hölder Spaces

Chapter 6. Hölder Estimates for the 1-dimensional Model Problems

Chapter 7. Hölder Estimates for Higher Dimensional Corner Models

Chapter 8. Hölder Estimates for Euclidean Models

Chapter 9. Hölder Estimates for General Models

Part III. Analysis of Generalized Kimura Diffusions

Chapter 10. Existence of Solutions

Chapter 11. The Resolvent Operator

Chapter 12. The Semi-group on ℂ°(P)

Appendix A: Proofs of Estimates for the Degenerate 1-d Model

Bibliography

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

9781400847181

1400847184

9781299051454

1299051456

9781400846108

1400846102

OCLC:

824353625