### Abstract

In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.

Original language | English (US) |
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Pages (from-to) | 1299-1322 |

Number of pages | 24 |

Journal | Electronic Journal of Probability |

Volume | 12 |

State | Published - Oct 19 2007 |

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### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Electronic Journal of Probability*,

*12*, 1299-1322.

**Ergodic properties of multidimensional Brownian motion with rebirth.** / Grigorescu, Ilie; Kang, Min.

Research output: Contribution to journal › Article

*Electronic Journal of Probability*, vol. 12, pp. 1299-1322.

}

TY - JOUR

T1 - Ergodic properties of multidimensional Brownian motion with rebirth

AU - Grigorescu, Ilie

AU - Kang, Min

PY - 2007/10/19

Y1 - 2007/10/19

N2 - In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.

AB - In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and determine explicitly the spectrum in dimension one. Two proofs of the exponential ergodicity are given, one using the inverse Laplace transform and properties of analytic semigroups, and the other based on Doeblin's condition. Both methods admit generalizations to a wide class of processes.

UR - http://www.scopus.com/inward/record.url?scp=35549000597&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35549000597&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:35549000597

VL - 12

SP - 1299

EP - 1322

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -