Ergodic Properties of Multidimensional Brownian Motion with Rebirth

Ilie Grigorescu (University of Miami)
Min Kang (North Carolina State University)

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.

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Pages: 1299-1322

Publication Date: October 19, 2007

DOI: 10.1214/EJP.v12-450

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