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Quasi-stationary distributions for a Brownian motion with drift and associated limit laws

Part of: Markov processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Servet Martinez  and
Jaime San Martin
Servet Martinez
Affiliation:
Universidad de Chile
Jaime San Martin*
Affiliation:
Universidad de Chile
*
∗Postal address: Departamento de Ingenieria Matemática, Facultad de Ciencias Fisicas y Matemáticas, Universidad de Chile, Casilla 170–3 Correo 3, Santiago, Chile. e-mail: [email protected], [email protected]
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Abstract

We prove that the quasi-invariant measures associated to a Brownian motion with negative drift X form a one-parameter family. The minimal one is a probability measure inducing the transition density of a three-dimensional Bessel process, and it is shown that it is the density of the limit distribution limt→∞Px(X A | τ > t). It is also shown that the minimal quasi-invariant measure of infinite mass induces the density of the limit distribution ) which is the law of a Bessel process with drift.

Keywords

BROWNIAN MOTION WITH DRIFTQUASI-STATIONARY DISTRIBUTIONS

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60J65: Brownian motion
Type
Research Article
Information
Journal of Applied Probability , Volume 31 , Issue 4 , December 1994 , pp. 911 - 920
Copyright
Copyright © Applied Probability Trust 1994 

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