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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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The Semi-group on

The Semi-group on

(p.235) Chapter TwelveThe Semi-group on %( p )
Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein

Rafe1 Mazzeo

Princeton University Press

This chapter deals with the semi-group on the space Β‎⁰(P). It first describes the boundary behavior of elements of the adjoint operator at points in the interiors of hypersurface boundary components before discussing the null-space of the adjoint under the hypothesis that a generalized Kimura diffusion operator, L, meets bP cleanly. It then examines long time asymptotics, along with a lemma in which P is a compact manifold with corners and L is a generalized Kimura diffusion on P. It also considers the existence of irregular solutions to the homogeneous equations Lu = f, for functions that do not belong to the range of the generator of a C⁰-semi-group on Β‎⁰(P).

Keywords:   semi-group, boundary behavior, adjoint operator, hypersurface boundary, null-space, generalized Kimura diffusion, Kimura diffusion operator, long time asymptotics, manifold with corners, irregular solution

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