Degenerate diffusion operators arising in population biology / Charles L. Epstein and Rafe Mazzeo.

Includes bibliographical references (pages 301-303) and index.

Introduction. Part 1: Wright-Fisher geometry and the maximum principle. Wright-Fisher geometry ; Maximum principles and uniqueness theorems. -- Part 2: Analysis of model problems. The model solution operators ; Degenerate H�eolder spaces ; H�eolder estimates for the 1-dimensional model problems ; H�eolder estimates for higher dimensional corner models ; H�eolder estimates for Euclidean models ; H�eolder estimates for general models. -- Part 3: Analysis of generalized Kimura diffusions. Existence of solutions ; The resolvent operator ; The semi-group on �A p0 s(P).

"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 H�eolder 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."--