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Limit motion of an Ornstein–Uhlenbeck particle on the equilibrium manifold of a force field

Part of: Markov processes Stochastic analysis

Published online by Cambridge University Press:  14 July 2016

Antonella Calzolari  and
Federico Marchetti
Antonella Calzolari*
Affiliation:
Università di Roma ‘Tor Vergata'
Federico Marchetti*
Affiliation:
Politecnico di Milano
*
∗Postal address: Dipartimento di Matematica, Università di Roma ‘Tor Vergata', via d. Ricerca Scientifica, 00133 Roma, Italy.
∗Postal address: Dipartimento di Matematica, Università di Roma ‘Tor Vergata', via d. Ricerca Scientifica, 00133 Roma, Italy.
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Abstract

In this paper we consider a position–velocity Ornstein-Uhlenbeck process in an external gradient force field pushing it toward a smoothly imbedded submanifold of . The force is chosen so that is asymptotically stable for the associated deterministic flow. We examine the asymptotic behavior of the system when the force intensity diverges together with the diffusion and the damping coefficients, with appropriate speed. We prove that, under some natural conditions on the initial data, the sequence of position processes is relatively compact, any limit process is constrained on , and satisfies an explicit stochastic differential equation which, for compact , has a unique solution.

Keywords

DIFFUSIONS ON MANIFOLDSORNSTEIN–UHLENBECK PARTICLECONSTRAINTSWEAK CONVERGENCECONVERGENCE OF STOCHASTIC INTEGRALS

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 924 - 938
Copyright
Copyright © Applied Probability Trust 1997 

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