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Degenerate Diffusion Operators Arising in Population Biology (AM-185)$
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Charles L. Epstein and Rafe Mazzeo

Print publication date: 2013

Print ISBN-13: 9780691157122

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691157122.001.0001

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Wright-Fisher Geometry

Wright-Fisher Geometry

(p.25) Chapter Two Wright-Fisher Geometry
Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein

Rafe1 Mazzeo

Princeton University Press

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural domains of definition for generalized Kimura diffusions: polyhedra in Euclidean space or, more generally, abstract manifolds with corners. Amongst the convex polyhedra, the chapter distinguishes the subclass of regular convex polyhedra P. P is a regular convex polyhedron if it is convex and if near any corner, P is the intersection of no more than N half-spaces with corresponding normal vectors that are linearly independent. These definitions establish that any regular convex polyhedron is a manifold with corners. The chapter concludes by defining the general class of elliptic Kimura operators on a manifold with corners P and shows that there is a local normal form for any operator L in this class.

Keywords:   generalized Kimura diffusion, Euclidean space, polyhedron, normal vector, manifold with corners, elliptic Kimura operator, normal form, Wright–Fisher geometry

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