320 pages, 3 b/w illustrations
Degenerate Diffusion Operators Arising in Population Biology 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 Degenerate Diffusion Operators Arising in Population Biology 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.
1 Introduction 1
2 Wright-Fisher Geometry 25
3 Maximum Principles and Uniqueness Theorems 34
4 The Model Solution Operators 51
5 Degenerate Holder Spaces 64
6 Holder Estimates for the 1-dimensional Model Problems 78
7 Holder Estimates for Higher Dimensional CornerModels 107
8 Holder Estimates for Euclidean Models 137
9 Holder Estimates for General Models 143
10 Existence of Solutions 181
11 The Resolvent Operator 218
12 The Semi-group on C0(P) 235
A1 Basic Kernel Estimates 252
A2 First Derivative Estimates 272
A3 Second Derivative Estimates 278
A4 Off-diagonal and Large-t Behavior 291
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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.