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Annals of Mathematics Studies #185: Degenerate Diffusion Operators Arising in Population Biologyby Charles L Epstein
Synopses & ReviewsPublisher 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 halfplane. Epstein and Mazzeo also demonstrate precise asymptotic results for the longtime behavior of solutions to both the forward and backward Kolmogorov equations. About the AuthorCharles 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 ContentsPreface xi
I WrightFisher Geometry and the Maximum Principle 23
3 Maximum Principles and Uniqueness Theorems 34
II Analysis of Model Problems 49
5 Degenerate Hölder Spaces 64
6 Hölder Estimates for the 1dimensional Model Problems 78
7 Hölder Estimates for Higher Dimensional CornerModels 107
8 Hölder Estimates for Euclidean Models 137
9 Hölder Estimates for General Models 143
III Analysis of Generalized Kimura Diffusions 179
11 The Resolvent Operator 218
12 The Semigroup on C0(P) 235
A Proofs of Estimates for the Degenerate 1d Model 251
Bibliography 301 What Our Readers Are SayingBe the first to add a comment for a chance to win!Product Details
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