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.
Mohammed S.E.A., Scheutzow M.: Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part II: examples and case studies. Ann. Prob. 25, 1210–1240 (1997)MATHCrossRefMathSciNetGoogle Scholar
Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)MATHGoogle Scholar
Reiß M., Riedle M., van Gaans O.: Delay differential equations driven by Lévy processes: stationarity and Feller properties. Stoch. Process. Appl. 116, 1409–1432 (2006)MATHCrossRefGoogle Scholar
van Renesse, M., Scheutzow, M.: Existence and uniqueness of solutions of stochastic functional differential equations. (2008, Submitted)Google Scholar
Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293. Springer-Verlag, Berlin (1991)Google Scholar
Rey-Bellet L., Thomas L.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225(2), 305–329 (2002)MATHCrossRefMathSciNetGoogle Scholar
Scheutzow M.: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12, 41–80 (1984)MATHMathSciNetGoogle Scholar