Seminars and Colloquia by Series

Friday, September 11, 2009 - 15:00 , Location: Skiles 154 , Sergio Almada , Georgia Tech , Organizer:
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.
Friday, March 27, 2009 - 16:00 , Location: Skiles 268 , Ricardo Restrepo , School of Mathematics, Georgia Tech , Organizer:
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).
Friday, March 6, 2009 - 15:00 , Location: Skiles 268 , Alex Grigo , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, February 27, 2009 - 15:00 , Location: Skiles 268 , Sergio Almada , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, February 20, 2009 - 15:00 , Location: Skiles 268 , Sergio Almada , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, February 13, 2009 - 15:00 , Location: Skiles 268 , Stas Minsker , School of Mathematics, Georgia Tech , Organizer:
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.
Friday, November 7, 2008 - 15:00 , Location: Skiles 168 , Yuri Bakhtin , School of Mathematics, Georgia Tech , Organizer:
A discrete infinite volume limit for random trees will be constructed and studied.
Friday, October 10, 2008 - 15:00 , Location: Skiles 168 , Stas Minsker , School of Mathematics, Georgia Tech , Organizer:
Based on a paper by E. Candes and Y. Plan.
Friday, October 3, 2008 - 15:00 , Location: Skiles 168 , Christian Houdre , School of Mathematics, Georgia Tech , Organizer:
Friday, September 26, 2008 - 15:00 , Location: Skiles 168 , Stas Minsker , School of Mathematics, Georgia Tech , Organizer:

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