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.

• 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 Holder Spaces 64*5.1 Standard Holder Spaces 65 *5.2 WF-Holder Spaces in 1-dimension 66 6 Holder Estimates for the 1-dimensional Model Problems 78*6.1 Kernel Estimates for Degenerate Model Problems 80 *6.2 Holder Estimates for the 1-dimensional Model Problems 89 *6.3 Propertiesof the Resolvent Operator 103 7 Holder 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 Holder Estimates for Euclidean Models 137*8.1 Holder Estimates for Solutions in the Euclidean Case 137 *8.2 1-dimensional Kernel Estimates 139 9 Holder 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-Holder 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

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.

• 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 Holder Spaces 64*5.1 Standard Holder Spaces 65 *5.2 WF-Holder Spaces in 1-dimension 66 6 Holder Estimates for the 1-dimensional Model Problems 78*6.1 Kernel Estimates for Degenerate Model Problems 80 *6.2 Holder Estimates for the 1-dimensional Model Problems 89 *6.3 Propertiesof the Resolvent Operator 103 7 Holder 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 Holder Estimates for Euclidean Models 137*8.1 Holder Estimates for Solutions in the Euclidean Case 137 *8.2 1-dimensional Kernel Estimates 139 9 Holder 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-Holder 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