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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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Degenerate Holder Spaces

Degenerate Holder Spaces

Chapter:
(p.64) Chapter Five Degenerate Hölder Spaces
Source:
Degenerate Diffusion Operators Arising in Population Biology (AM-185)
Author(s):

Charles L. Epstein

Rafe1 Mazzeo

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691157122.003.0005

This chapter introduces the degenerate Hölder spaces on the spaces Sₙ,ₘ, and their heat-space counterparts on Sₙ,ₘ x [0, T]. It also establishes the basic properties of these spaces. The objective is to seek function spaces on the domain P for which the diffusion associated to a generalized Kimura diffusion operator L is well posed. More pragmatically, the chapter wishes to define spaces on which one can prove analogues of the standard parabolic Schauder estimates in order to pass from the model to more general operators. The chapter first considers the classical interior Hölder spaces and their parabolically scaled “heat” analogues before discussing Wright–Fisher–Hölder spaces on 1-dimension. It also defines analogues of all the spaces above which have k full derivatives in the x direction: these are the hybrid spaces.

Keywords:   hybrid space, Hölder space, Schauder estimate, function space, generalized Kimura diffusion, Kimura diffusion operator

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