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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 Model Solution Operators

The Model Solution Operators

(p.51) Chapter Four The Model Solution Operators
Degenerate Diffusion Operators Arising in Population Biology (AM-185)

Charles L. Epstein

Rafe1 Mazzeo

Princeton University Press

This chapter introduces the model problems and the solution operator for the associated heat equations. These operators give a good approximation for the behavior of the heat kernel in neighborhoods of different types of boundary points. The chapter states and proves the elementary features of these operators and shows that the model heat operators have an analytic continuation to the right half plane. It first considers the model problem in 1-dimension and in higher dimensions before discussing the solution to the homogeneous Cauchy problem. It then describes the first steps toward perturbation theory and constructs the solution operator for generalized Kimura diffusions on a suitable scale of Hölder spaces. It also defines the resolvent families and explains why the estimates obtained here are not adequate for the perturbation theoretic arguments needed to construct the solution operator for generalized Kimura diffusions.

Keywords:   model problem, solution operator, heat equation, heat kernel, homogeneous Cauchy problem, perturbation theory, generalized Kimura diffusion, Hölder space

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