Degenerate Diffusion Operators Arising in Population Biology (AM-185) ：Degenerate Diffusion Operators Arising in Population Biology (AM-185) ( Annals of Mathematics Studies )

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

Publication series ：Annals of Mathematics Studies

Author： Epstein Charles L.;Mazzeo Rafe;;  

Publisher： Princeton University Press‎

Publication year： 2013

E-ISBN: 9781400846108

P-ISBN(Paperback): 9780691157122

Subject： O175.3 The differential operator theory

Keyword： 普通生物学,数理科学和化学,数学

Language： ENG

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Description

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.

Chapter

Cover

Annals of Mathematics Studies Number 185

Title

Copyright

Dedication

Contents

Preface

1 Introduction

1.1 Generalized Kimura Diffusions

1.2 Model Problems

1.3 Perturbation Theory

1.4 Main Results

1.5 Applications in Probability Theory

1.6 Alternate Approaches

1.7 Outline of Text

1.8 Notational Conventions

I Wright-Fisher Geometry and the Maximum Principle

2 Wright-Fisher Geometry

2.1 Polyhedra and Manifolds with Corners

2.2 Normal Forms and Wright-Fisher Geometry

3 Maximum Principles and Uniqueness Theorems

3.1 Model Problems

3.2 Kimura Diffusion Operators on Manifolds with Corners

3.3 Maximum Principles for the Heat Equation

II Analysis of Model Problems

4 The Model Solution Operators

4.1 The Model Problem in 1-dimension

4.2 The Model Problem in Higher Dimensions

4.3 Holomorphic Extension

4.4 First Steps Toward Perturbation Theory

5 Degenerate Hölder Spaces

5.1 Standard Hölder Spaces

5.2 WF-Hölder Spaces in 1-dimension

6 Hölder Estimates for the 1-dimensional Model Problems

6.1 Kernel Estimates for Degenerate Model Problems

6.2 Hölder Estimates for the 1-dimensional Model Problems

6.3 Properties of the Resolvent Operator

7 Hölder Estimates for Higher Dimensional Corner Models

7.1 The Cauchy Problem

7.2 The Inhomogeneous Case

7.3 The Resolvent Operator

8 Hölder Estimates for Euclidean Models

8.1 Hölder Estimates for Solutions in the Euclidean Case

8.2 1-dimensional Kernel Estimates

9 Hölder Estimates for General Models

9.1 The Cauchy Problem

9.2 The Inhomogeneous Problem

9.3 Off-diagonal and Long-time Behavior

9.4 The Resolvent Operator

III Analysis of Generalized Kimura Diffusions

10 Existence of Solutions

10.1 WF-Hölder Spaces on a Manifold with Corners

10.2 Overview of the Proof

10.3 The Induction Argument

10.4 The Boundary Parametrix Construction

10.5 Solution of the Homogeneous Problem

10.6 Proof of the Doubling Theorem

10.7 The Resolvent Operator and C0-Semi-group

10.8 Higher Order Regularity

11 The Resolvent Operator

11.1 Construction of the Resolvent

11.2 Holomorphic Semi-groups

11.3 Diffusions Where All Coefficients Have the Same Leading Homogeneity

12 The Semi-group on L0(P)

12.1 The Domain of the Adjoint

12.2 The Null-space of L

12.3 Long Time Asymptotics

12.4 Irregular Solutions of the Inhomogeneous Equation

A Proofs of Estimates for the Degenerate 1-d Model

A.1 Basic Kernel Estimates

A.2 First Derivative Estimates

A.3 Second Derivative Estimates

A.4 Off-diagonal and Large-t Behavior

Bibliography

Index

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