Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein; Rafe Mazzeo

Published by Princeton University Press 2013

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein and Rafe Mazzeo

Volume 185 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400846108

Overview
Contents

About this book

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 codimensional 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.

Author / Editor information

Charles L. Epstein is the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania. Rafe Mazzeo is professor of mathematics at Stanford University.

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	vii
Preface Requires Authentication Unlicensed Licensed Download PDF	xi
Chapter 1. Introduction Requires Authentication Unlicensed Licensed Download PDF	1
Part I. Wright-Fisher Geometry and the Maximum Principle
Chapter 2. Wright-Fisher Geometry Requires Authentication Unlicensed Licensed Download PDF	25
Chapter 3. Maximum Principles and Uniqueness Theorems Requires Authentication Unlicensed Licensed Download PDF	34
Part II. Analysis of Model Problems
Chapter 4. The Model Solution Operators Requires Authentication Unlicensed Licensed Download PDF	51
Chapter 5. Degenerate Hölder Spaces Requires Authentication Unlicensed Licensed Download PDF	64
Chapter 6. Hölder Estimates for the 1-dimensional Model Problems Requires Authentication Unlicensed Licensed Download PDF	78
Chapter 7. Hölder Estimates for Higher Dimensional Corner Models Requires Authentication Unlicensed Licensed Download PDF	107
Chapter 8. Hölder Estimates for Euclidean Models Requires Authentication Unlicensed Licensed Download PDF	137
Chapter 9. Hölder Estimates for General Models Requires Authentication Unlicensed Licensed Download PDF	143
Part III. Analysis of Generalized Kimura Diffusions
Chapter 10. Existence of Solutions Requires Authentication Unlicensed Licensed Download PDF	181
Chapter 11. The Resolvent Operator Requires Authentication Unlicensed Licensed Download PDF	218
Chapter 12. The Semi-group on ℂ°(P) Requires Authentication Unlicensed Licensed Download PDF	235
Appendix A: Proofs of Estimates for the Degenerate 1-d Model Requires Authentication Unlicensed Licensed Download PDF	251
Bibliography Requires Authentication Unlicensed Licensed Download PDF	301
Index Requires Authentication Unlicensed Licensed Download PDF	305

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