Murakami Sale
 
 

Recently Viewed clear list


Original Essays | August 18, 2014

Ian Leslie: IMG Empathic Curiosity



Today, we wonder anxiously if digital media is changing our brains. But if there's any time in history when our mental operations changed... Continue »
  1. $18.89 Sale Hardcover add to wish list

spacer
Qualifying orders ship free.
$75.00
New Trade Paper
Ships in 1 to 3 days
Add to Wishlist
available for shipping or prepaid pickup only
Available for In-store Pickup
in 7 to 12 days
Qty Store Section
2 Remote Warehouse Mathematics- Differential Equations

More copies of this ISBN

This title in other editions

Annals of Mathematics Studies #185: Degenerate Diffusion Operators Arising in Population Biology

by

Annals of Mathematics Studies #185: Degenerate Diffusion Operators Arising in Population Biology Cover

 

Synopses & Reviews

Publisher Comments:

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.

About the Author

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.

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

Product Details

ISBN:
9780691157153
Author:
Epstein, Charles L.
Publisher:
Princeton University Press
Author:
Mazzeo, Rafe
Subject:
Differential Equations
Subject:
Biological Sciences.
Subject:
Mathematics
Subject:
Environmental Studies-General
Copyright:
Edition Description:
Trade paper
Publication Date:
20130431
Binding:
TRADE PAPER
Language:
English
Illustrations:
3 line illus.
Pages:
320
Dimensions:
9 x 6 in

Related Subjects

Science and Mathematics » Biology » General
Science and Mathematics » Environmental Studies » General
Science and Mathematics » Mathematics » Biological Sciences
Science and Mathematics » Mathematics » Differential Equations

Annals of Mathematics Studies #185: Degenerate Diffusion Operators Arising in Population Biology New Trade Paper
0 stars - 0 reviews
$75.00 In Stock
Product details 320 pages Princeton University Press - English 9780691157153 Reviews:
spacer
spacer
  • back to top
Follow us on...




Powell's City of Books is an independent bookstore in Portland, Oregon, that fills a whole city block with more than a million new, used, and out of print books. Shop those shelves — plus literally millions more books, DVDs, and gifts — here at Powells.com.