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Brownian Motion on the Figure Eight

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Abstract

In an interval containing the origin we study a Brownian motion which returns to zero as soon as it reaches the boundary. We determine explicitly its transition probability, prove it is ergodic and calculate the decay rate to equilibrium. It is shown that the process solves the martingale problem for certain asymmetric boundary conditions and can be regarded as a diffusion on an eight shaped domain. In the case the origin is situated at a rationally commensurable distance from the two endpoints of the interval we give the complete characterization of the possibility of collapse of distinct paths.

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Authors and Affiliations

  1. Department of Mathematics, University of Miami, 1365 Memorial Dr., Room 515, Coral Gables, Florida, 33143

    Ilie Grigorescu

  2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois, 60208-2730

    Min Kang

Authors
  1. Ilie Grigorescu
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  2. Min Kang
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Correspondence to Ilie Grigorescu.

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Grigorescu, I., Kang, M. Brownian Motion on the Figure Eight. Journal of Theoretical Probability 15, 817–844 (2002). https://doi.org/10.1023/A:1016232201962

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  • DOI: https://doi.org/10.1023/A:1016232201962

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