Ergodic properties of multidimensional Brownian motion with rebirth

Ilie Grigorescu, Min Kang

Research output: Contribution to journalArticle

14 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1299-1322
Number of pages24
JournalElectronic Journal of Probability
Volume12
StatePublished - Oct 19 2007

Fingerprint

Exponential Ergodicity
Inverse Laplace Transform
Dirichlet Laplacian
Analytic Semigroup
Interior Point
Resolvent
Spectral Properties
One Dimension
Brownian motion
Modulo
Green's function
Closed-form
Fixed point
Calculate
Invariant
Class
Generalization

ASJC Scopus subject areas

  • Statistics and Probability

Cite this

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

In: Electronic Journal of Probability, Vol. 12, 19.10.2007, p. 1299-1322.

Research output: Contribution to journalArticle

@article{8aa529b18d094928b6bc1c54b9b2c212,
title = "Ergodic properties of multidimensional Brownian motion with rebirth",
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.",
author = "Ilie Grigorescu and Min Kang",
year = "2007",
month = "10",
day = "19",
language = "English (US)",
volume = "12",
pages = "1299--1322",
journal = "Electronic Journal of Probability",
issn = "1083-6489",
publisher = "Institute of Mathematical Statistics",

}

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 -