### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 817-844 |

Number of pages | 28 |

Journal | Journal of Theoretical Probability |

Volume | 15 |

Issue number | 3 |

DOIs | |

State | Published - 2002 |

### Fingerprint

### Keywords

- Absorbing Brownian motion
- Decay rate
- Ergodicity
- Laplace transform

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Journal of Theoretical Probability*,

*15*(3), 817-844. https://doi.org/10.1023/A:1016232201962

**Brownian motion on the figure eight.** / Grigorescu, Ilie; Kang, Min.

Research output: Contribution to journal › Article

*Journal of Theoretical Probability*, vol. 15, no. 3, pp. 817-844. https://doi.org/10.1023/A:1016232201962

}

TY - JOUR

T1 - Brownian motion on the figure eight

AU - Grigorescu, Ilie

AU - Kang, Min

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - Absorbing Brownian motion

KW - Decay rate

KW - Ergodicity

KW - Laplace transform

UR - http://www.scopus.com/inward/record.url?scp=0141464960&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141464960&partnerID=8YFLogxK

U2 - 10.1023/A:1016232201962

DO - 10.1023/A:1016232201962

M3 - Article

AN - SCOPUS:0141464960

VL - 15

SP - 817

EP - 844

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

IS - 3

ER -