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Dynamics of Morse-Smale urn processes

Published online by Cambridge University Press:  14 October 2010

Michel Benaïm  and
Morris W. Hirsch
Michel Benaïm
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720 USA
Morris W. Hirsch
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720 USA
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Abstract

We consider stochastic processes {xn}n≥0 of the form

where F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1005 - 1030
Copyright
Copyright © Cambridge University Press 1995

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