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Analysis of Degenerate Diffusion Operators Arising in Population Biology

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From Fourier Analysis and Number Theory to Radon Transforms and Geometry

Part of the book series: Developments in Mathematics ((DEVM,volume 28))

Abstract

In this chapter, we describe our recent work on the analytic foundations for the study of degenerate diffusion equations which arise as the infinite population limits of standard models in population genetics. Our principal results concern existence, uniqueness, and regularity of solutions when the data belong to anisotropic Hölder spaces, adapted to the degeneracy of these operators. These results suffice to prove the existence of a strongly continuous \({\mathcal{C}}^{0}\)-semigroup. The details of the definitions and complete proofs of these results can be found in [8].

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References

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Acknowledgements

We would like to acknowledge the generous financial and unflagging personal support provided by Ben Mann and the DARPA FunBio project. It is certainly the case that without Ben’s encouragement, we would never have undertaken this project. We would like to thank our FunBio colleagues who provided us with the motivation and knowledge base to pursue this project and Simon Levin for his leadership and inspiration. CLE would like to thank Warren Ewens, Josh Plotkin, and Ricky Der, from whom he has learned most of what he knows about population genetics. We would both like to thank Charlie Fefferman for showing us an explicit formula for \({k}_{t}^{0}(x,y),\) which set us off in the very fruitful direction pursued herein. Work of Charles L. Epstein research partially supported by NSF grant DMS06-03973, and DARPA grants HR00110510057 and HR00110910055. Work of Rafe Mazzeo research partially supported by NSF grant DMS08-05529, and DARPA grants HR00110510057 and HR00110910055. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors, and do not nessarily, reflect the views of either the National Science Foundation or DARPA.

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Correspondence to Charles L. Epstein .

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This paper is dedicated to the memory of Leon Ehrenpreis, a giant in the field of partial differential equations

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Epstein, C.L., Mazzeo, R. (2013). Analysis of Degenerate Diffusion Operators Arising in Population Biology. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_8

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