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

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

8 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
Externally published	Yes

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

Access

10.1007/s00440-018-0840-2

OpenUrl availability

Full text

Cite this

@article{1bb5f47ea00e40a0bb76832c4ed8657e,

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.",

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

author = "Epstein, {Charles L.} and Pop, {Camelia A.}",

note = "Funding Information: C. L. Epstein{\textquoteright}s research is partially supported by NSF Grant DMS-1507396. C.A. Pop{\textquoteright}s research is partially supported by NSF Grant DMS-1714490. Publisher Copyright: {\textcopyright} 2018, Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2019",

month = feb,

day = "4",

doi = "10.1007/s00440-018-0840-2",

language = "English (US)",

volume = "173",

pages = "537--603",

journal = "Probability Theory and Related Fields",

issn = "0178-8051",

publisher = "Springer New York",

number = "1-2",

}

TY - JOUR

T1 - Transition probabilities for degenerate diffusions arising in population genetics

AU - Epstein, Charles L.

AU - Pop, Camelia A.

N1 - 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.

PY - 2019/2/4

Y1 - 2019/2/4

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

UR - http://www.scopus.com/inward/record.url?scp=85042614407&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85042614407&partnerID=8YFLogxK

U2 - 10.1007/s00440-018-0840-2

DO - 10.1007/s00440-018-0840-2

M3 - Article

AN - SCOPUS:85042614407

SN - 0178-8051

VL - 173

SP - 537

EP - 603

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 1-2

ER -

Transition probabilities for degenerate diffusions arising in population genetics

Abstract

Bibliographical note

Keywords

Access

OpenUrl availability

Other files and links

Fingerprint

Cite this