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 [EMS01, Mat02b, Hai02, BM05] on which this work builds. As in [BM05] 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.
In this article, we study the approximation of a probability measure µ on R d by its empirical measureμN interpreted as a random quantization. As error criterion we consider an averaged p-th moment Wasserstein metric. In the case where 2p < d, we establish fine upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.
Under rather general conditions we show that any monotone random dynamical system on an (admissible) subset of a partially ordered Banach space V has a unique invariant measure. This measure is Dirac, i.e. it is generated by some stationary process. If the cone V þ of non-negative elements of V is normal, then this stationary process is a global random attractor with respect to convergence in probability. As examples we consider one-dimensional ordinary and retarded stochastic differential equations, a stochastic model of a biochemical control circuit, a class of parabolic stochastic partial differential equations (PDEs) with additive noise and interacting particle systems.
27 pagesInternational audienceWe analyze common lifts of stochastic processes to rough paths/rough drivers-valued processes and give sufficient conditions for the cocycle property to hold for these lifts. We show that random rough differential equations driven by such lifts induce random dynamical systems. In particular, our results imply that rough differential equations driven by the lift of fractional Brownian motion in the sense of Friz-Victoir induce random dynamical systems
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