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Ergodic Properties of Multidimensional Brownian Motion with Rebirth
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Ilie Grigorescu, University of Miami
Min Kang, North Carolina State University
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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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Full text: PDF
Pages: 1299-1322
Published on: October 19, 2007
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Electronic Journal of Probability. ISSN: 1083-6489 |
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