Transition probabilities for degenerate diffusions arising in population genetics

Charles L. Epstein; Camelia A. Pop

doi:10.1007/s00440-018-0840-2

Transition probabilities for degenerate diffusions arising in population genetics

Charles L. Epstein, Camelia A. Pop

School of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.

Original language	English (US)
Pages (from-to)	537-603
Number of pages	67
Journal	Probability Theory and Related Fields
Volume	173
Issue number	1-2
DOIs	https://doi.org/10.1007/s00440-018-0840-2
State	Published - Feb 4 2019

Bibliographical note

Funding Information:
C. L. Epstein’s research is partially supported by NSF Grant DMS-1507396. C.A. Pop’s research is partially supported by NSF Grant DMS-1714490.

Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

Caloric measure
Compact manifold with corners
Degenerate elliptic operators
Dirichlet heat kernel
Fundamental solution
Hitting distributions
Markov processes
Transition probabilities

Publisher link

10.1007/s00440-018-0840-2

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title = "Transition probabilities for degenerate diffusions arising in population genetics",

abstract = "We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.",

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AU - Pop, Camelia A.

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N2 - We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.

AB - We provide a detailed description of the structure of the transition probabilities and of the hitting distributions on boundary components of a manifold with corners for a degenerate strong Markov process arising in population genetics. The Markov processes that we study are a generalization of the classical Wright–Fisher process. The main ingredients in our proofs are based on the analysis of the regularity properties of solutions to a forward Kolmogorov equation defined on a compact manifold with corners, which is degenerate in the sense that it is not strictly elliptic and the coefficients of the first order drift term have mild logarithmic singularities.

KW - Caloric measure

KW - Compact manifold with corners

KW - Degenerate elliptic operators

KW - Dirichlet heat kernel

KW - Fundamental solution

KW - Hitting distributions

KW - Markov processes

KW - Transition probabilities

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Transition probabilities for degenerate diffusions arising in population genetics

Abstract

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