Degenerate Diffusion Operators Arising in Population Biology

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