Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Degenerate Diffusion Operators Arising in Population Biology (AM-185)

by Charles L. Epstein, Rafe Mazzeo
     
 

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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

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Overview

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.

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Product Details

ISBN-13:
9781400846108
Publisher:
Princeton University Press
Publication date:
04/04/2013
Series:
Annals of Mathematics Studies
Sold by:
Barnes & Noble
Format:
NOOK Book
Pages:
320
File size:
24 MB
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Meet 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.

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