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Series: Probability Working Seminar

The talk is based on the recent paper by M.Hairer, J.Mattingly, and M.Scheutzow with the same title.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.

Series: Probability Working Seminar

This talk is based in the article titled "On the convergence to equilibrium of Kac’s random walk on matrices" by Roberto Oliveira (IMPA, Brazil). We show how a strategy related to the path coupling method allows us to establish tight bounds for the L-2 transportation-cost mixing time of the Kac's random walk on SO(n).

Series: Probability Working Seminar

The talk is based on the paper titled "Anosov diffeomorphisms and coupling" by Bressaud and Liverani. Existence and uniqueness of SRB invariant measure for the dynamics is established via a coupling of initial conditions introduced to dynamics by L.-S. Young.

Series: Probability Working Seminar

This is a continuation of last week's seminar. The talk is based on a paper by Kuksin, Pyatnickiy, and Shirikyan. In this paper, the convergence to a stationary distribution is established by partial coupling. Here, only finitely many coordinates in the (infinite-dimensional) phase space participate in the coupling while the dynamics takes care of the other coordinates.

Series: Probability Working Seminar

The talk is based on a paper by Kuksin, Pyatnickiy, and Shirikyan. In this paper, the convergence to a stationary distribution is established by partial coupling. Here, only finitely many coordinates in the (infinite-dimensional) phase space participate in the coupling while the dynamics takes care of the other coordinates.

Series: Probability Working Seminar

This term, the main topic for the Probability Working Seminar will be the coupling method, broadly understood. In the first talk, some basics on coupling will be discussed along with classical examples such as the ergodic theorem for Markov chains.

Series: Probability Working Seminar

A discrete infinite volume limit for random trees will be constructed and studied.

Series: Probability Working Seminar

Based on a paper by E. Candes and Y. Plan.

Series: Probability Working Seminar

Series: Probability Working Seminar