Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations

Loading...
Thumbnail Image

Date

2011-03-01

Journal Title

Journal ISSN

Volume Title

Repository Usage Stats

219
views
263
downloads

Citation Stats

Abstract

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris' theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such "asymptotic couplings" were central to (Mattingly and Sinai in Comm Math Phys 219(3):523-565, 2001; Mattingly in Comm Math Phys 230(3):461-462, 2002; Hairer in Prob Theory Relat Field 124:345-380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553-582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris' celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are "small" (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a "small set" by the much weaker notion of a "d-small set," which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it. © 2009 Springer-Verlag.

Department

Description

Provenance

Subjects

Citation

Published Version (Please cite this version)

10.1007/s00440-009-0250-6

Publication Info

Hairer, M, JC Mattingly and M Scheutzow (2011). Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations. Probability Theory and Related Fields, 149(1). pp. 223–259. 10.1007/s00440-009-0250-6 Retrieved from https://hdl.handle.net/10161/10831.

This is constructed from limited available data and may be imprecise. To cite this article, please review & use the official citation provided by the journal.


Unless otherwise indicated, scholarly articles published by Duke faculty members are made available here with a CC-BY-NC (Creative Commons Attribution Non-Commercial) license, as enabled by the Duke Open Access Policy. If you wish to use the materials in ways not already permitted under CC-BY-NC, please consult the copyright owner. Other materials are made available here through the author’s grant of a non-exclusive license to make their work openly accessible.