UNSPECIFIED (2002) Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 101 (2). pp. 185-232. ISSN 0304-4149

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Abstract

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; for pathwise approximations it is, in general, only inherited by specialty constructed implicit discretizations. Examples of such discretization based on backward Enter methods are given, and approximation of the Langevin equation studied in some detail. (C) 2002 Elsevier Science B.V. All rights reserved.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Journal or Publication Title:	STOCHASTIC PROCESSES AND THEIR APPLICATIONS
Publisher:	ELSEVIER SCIENCE BV
ISSN:	0304-4149
Date:	October 2002
Volume:	101
Number:	2
Number of Pages:	48
Page Range:	pp. 185-232
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/10528

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