Open Access
2007 Ergodic Properties of Multidimensional Brownian Motion with Rebirth
Ilie Grigorescu, Min Kang
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Electron. J. Probab. 12: 1299-1322 (2007). DOI: 10.1214/EJP.v12-450


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.


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Ilie Grigorescu. Min Kang. "Ergodic Properties of Multidimensional Brownian Motion with Rebirth." Electron. J. Probab. 12 1299 - 1322, 2007.


Accepted: 19 October 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60073
MathSciNet: MR2346513
Digital Object Identifier: 10.1214/EJP.v12-450

Primary: 60J35
Secondary: 35K15 , 60J75 , 91B24 , 92D10

Keywords: Analytic semigroup , Dirichlet Laplacian , Green function , jump diffusion

Vol.12 • 2007
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