Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T07:11:00.310Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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