Description
Book SynopsisThis 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 a
Table of ContentsPreface 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
