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Degenerate Diffusion Operators Arising in Population Biology (AM-185)
Charles L. Epstein & Rafe Mazzeo

Book Description
Chapter 1 [in PDF format]

TABLE OF CONTENTS:

Preface xi
1 Introduction 1

  • 1.1 Generalized Kimura Diffusions 3
  • 1.2 Model Problems 5
  • 1.3 Perturbation Theory 9
  • 1.4 Main Results 10
  • 1.5 Applications in Probability Theory 13
  • 1.6 Alternate Approaches 14
  • 1.7 Outline of Text 16
  • 1.8 Notational Conventions 20

I Wright-Fisher Geometry and the Maximum Principle 23
2 Wright-Fisher Geometry 25

  • 2.1 Polyhedra and Manifolds with Corners 25
  • 2.2 Normal Forms and Wright-Fisher Geometry 29

3 Maximum Principles and Uniqueness Theorems 34

  • 3.1 Model Problems 34
  • 3.2 Kimura Diffusion Operators on Manifolds with Corners 35
  • 3.3 Maximum Principles for theHeat Equation 45

II Analysis of Model Problems 49
4 The Model Solution Operators 51

  • 4.1 The Model Problemin 1-dimension 51
  • 4.2 The Model Problem in Higher Dimensions 54
  • 4.3 Holomorphic Extension 59
  • 4.4 First Steps Toward Perturbation Theory 62

5 Degenerate Hölder Spaces 64

  • 5.1 Standard Hölder Spaces 65
  • 5.2 WF-Hölder Spaces in 1-dimension 66

6 Hölder Estimates for the 1-dimensional Model Problems 78

  • 6.1 Kernel Estimates for Degenerate Model Problems 80
  • 6.2 Hölder Estimates for the 1-dimensional Model Problems 89
  • 6.3 Propertiesof the Resolvent Operator 103

7 Hölder Estimates for Higher Dimensional CornerModels 107

  • 7.1 The Cauchy Problem 109
  • 7.2 The Inhomogeneous Case 122
  • 7.3 The Resolvent Operator 135

8 Hölder Estimates for Euclidean Models 137

  • 8.1 Hölder Estimates for Solutions in the Euclidean Case 137
  • 8.2 1-dimensional Kernel Estimates 139

9 Hölder Estimates for General Models 143

  • 9.1 The Cauchy Problem 145
  • 9.2 The Inhomogeneous Problem 149
  • 9.3 Off-diagonal and Long-time Behavior 166
  • 9.4 The Resolvent Operator 169

III Analysis of Generalized Kimura Diffusions 179
10 Existence of Solutions 181

  • 10.1 WF-Hölder Spaces on a Manifold with Corners 182
  • 10.2 Overview of the Proof 187
  • 10.3 The Induction Argument 191
  • 10.4 The Boundary Parametrix Construction 194
  • 10.5 Solution of the Homogeneous Problem 205
  • 10.6 Proof of the Doubling Theorem 208
  • 10.7 The Resolvent Operator and C0-Semi-group 209
  • 10.8 Higher Order Regularity 211

11 The Resolvent Operator 218

  • 11.1 Construction of the Resolvent 220
  • 11.2 Holomorphic Semi-groups 229
  • 11.3 DiffusionsWhere All Coefficients Have the Same Leading Homogeneity 230

12 The Semi-group on C0(P) 235

  • 12.1 The Domain of the Adjoint 237
  • 12.2 The Null-space of L 240
  • 12.3 Long Time Asymptotics 243
  • 12.4 Irregular Solutions of the Inhomogeneous Equation 247

A Proofs of Estimates for the Degenerate 1-d Model 251

  • A.1 Basic Kernel Estimates 252
  • A.2 First Derivative Estimates 272
  • A.3 Second Derivative Estimates 278
  • A.4 Off-diagonal and Large-t Behavior 291

Bibliography 301
Index 305

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File created: 4/17/2014

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