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Brownian Motion on the Figure Eight

Author

Listed:
  • Ilie Grigorescu

    (University of Miami)

  • Min Kang

    (Northwestern University)

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.

Suggested Citation

  • Ilie Grigorescu & Min Kang, 2002. "Brownian Motion on the Figure Eight," Journal of Theoretical Probability, Springer, vol. 15(3), pages 817-844, July.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:3:d:10.1023_a:1016232201962
    DOI: 10.1023/A:1016232201962
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    Cited by:

    1. Ilie Grigorescu & Min Kang, 2003. "Path Collapse for an Inhomogeneous Random Walk," Journal of Theoretical Probability, Springer, vol. 16(1), pages 147-159, January.

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