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

Part of: Limit theorems Markov processes

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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References

[1] Cavender, J. A. (1978) Quasi-stationary distributions of birth and death processes. Adv. Appl. Prob. 10, 570–586.CrossRefGoogle Scholar
[2] Dellacherie, C. and Meyer, P. A. (1987) Probabilités et potential, Théorie du potential associée à une resolvante. Théorie des processus de Markov. Hermann, Paris.Google Scholar
[3] Durrett, R. (1984) Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA.Google Scholar
[4] Ferrari, P., Martinez, S. and Picco, P. (1992) Existence of quasi-stationary distributions for birth-death chains. Adv. Appl. Prob. 19, 795–813.CrossRefGoogle Scholar
[5] Karatzas, I. and Shreve, S. E. (1988) Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.CrossRefGoogle Scholar
[6] Keener, R. W. (1992) Limit theorems for random walks conditioned to stay positive. Ann. Prob. 20, 801–824.CrossRefGoogle Scholar
[7] Knight, F. B. (1969) Brownian local times and taboo processes. Trans. Amer. Math. Soc. 143, 173–185.CrossRefGoogle Scholar
[8] Knight, F. B. (1981) Essentials of Brownian motion and diffusion. Math. Surveys 18, American Mathematical Society, Providence. RI.Google Scholar
[9] Rogers, L. and Pitman, J. (1981) Markov functions. Ann. Prob. 9, 573–582.CrossRefGoogle Scholar
[10] Seneta, E. and Vere-Jones, D. (1966) On quasi stationary distributions in discrete time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 404–434.CrossRefGoogle Scholar
[11] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford 13(2), 7–28.CrossRefGoogle Scholar
[12] Williams, D. (1970) Decomposing the Brownian path. Bull. Amer. Math. Soc. 76, 871–873.CrossRefGoogle Scholar