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Gene flow across geographical barriers — scaling limits of random walks with obstacles

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  • Forien, Raphaël

Abstract

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.

Suggested Citation

  • Forien, Raphaël, 2019. "Gene flow across geographical barriers — scaling limits of random walks with obstacles," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 3748-3773.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:10:p:3748-3773
    DOI: 10.1016/j.spa.2018.10.006
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    References listed on IDEAS

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    1. Nagylaki, Thomas, 2016. "Clines with partial panmixia across a geographical barrier," Theoretical Population Biology, Elsevier, vol. 109(C), pages 28-43.
    2. Mandrekar, Vidyadhar & Pilipenko, Andrey, 2016. "On a Brownian motion with a hard membrane," Statistics & Probability Letters, Elsevier, vol. 113(C), pages 62-70.
    3. Nagylaki, Thomas, 2012. "Clines with partial panmixia," Theoretical Population Biology, Elsevier, vol. 81(1), pages 45-68.
    4. Nagylaki, Thomas & Zeng, Kai, 2016. "Clines with partial panmixia across a geographical barrier in an environmental pocket," Theoretical Population Biology, Elsevier, vol. 110(C), pages 1-11.
    5. Nagylaki, Thomas, 2012. "Clines with partial panmixia in an unbounded unidimensional habitat," Theoretical Population Biology, Elsevier, vol. 82(1), pages 22-28.
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