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Asymptotic coupling and a weak form of Harris' theorem with applications to stochastic delay equations | | Martin Hairer
; Jonathan C. Mattingly
; Michael Scheutzow
; | | Date: |
26 Feb 2009 | | 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 recent work on SPDEs on which this work builds. 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. | | Source: | arXiv, 0902.4495 | | Services: | Forum | Review | PDF | Favorites | |
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