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Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations
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Hairer, Martin, Mattingly, Jonathan Christopher and Scheutzow, Michael (2011) Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probability Theory and Related Fields, Volume 149 (Numbers 12). pp. 223259. doi:10.1007/s0044000902506
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Official URL: http://dx.doi.org/10.1007/s0044000902506
Abstract
There are many Markov chains on infinite dimensional spaces whose onestep 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):523565, 2001; Mattingly in Comm Math Phys 230(3):461462, 2002; Hairer in Prob Theory Relat Field 124:345380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553582, 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 "dsmall set," which takes the topology of the underlying space into account via a distancelike function d. With this notion at hand, we prove an analogue to Harris' theorem, where the convergence takes place in a Wassersteinlike 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 longstanding open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.
Item Type:  Journal Article  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of Science > Mathematics  
Library of Congress Subject Headings (LCSH):  Stochastic differential equations, Invariant measures, Asymptotic expansions, Spectral theory (Mathematics)  
Journal or Publication Title:  Probability Theory and Related Fields  
Publisher:  Springer  
ISSN:  01788051  
Official Date:  February 2011  
Dates: 


Volume:  Volume 149  
Number:  Numbers 12  
Page Range:  pp. 223259  
DOI:  10.1007/s0044000902506  
Status:  Peer Reviewed  
Publication Status:  Published  
Access rights to Published version:  Restricted or Subscription Access  
Funder:  Engineering and Physical Sciences Research Council (EPSRC), National Science Foundation (U.S.) (NSF), Alfred P. Sloan Foundation, Deutsche Forschungsgemeinschaft (DFG)  
Grant number:  EP/D071593/1 (EPSRC), DMS0449910 (NSF), Forschergruppe 718 (DFG) 
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