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Series: Probability Working Seminar

We will discuss the role that self-avoiding walks play in sampling 'physical' models on graphs, allowing to translate the complicated calculation of the marginals to a tree recurrence which, under the appropriate conditions (e.g. some form of 'spatial mixing'), reduces to a polynomial recurrence. This talk is mainly based on Dror Weitz' article "Counting independent sets up to the tree threshold".

Series: Probability Working Seminar

We will present probability inequalities for the sums of independent, selfadjoint random matrices. The focus is made on noncommutative generalizations of the classical bounds of Azuma, Bernstein, Cherno ff, Hoeffding, among others. These inequalities imply concentration results for the empirical covariance matrices. No preliminary knowledge of probability theory will be assumed. (The talk is based on a paper by J. Tropp).

Series: Probability Working Seminar

We consider the small noise perturbation (in the Ito sense) of a one dimensional ODE. We study the case in which the ODE has not unique solution, but the SDE does. A particular setting of this sort is studied and the properties of the solution are obtained when the noise level vanishes. We relate this to give an example of a 1-dimensional transport equation without uniqueness of weak solution. We show how by a suitable random noise perturbation, the stochastic equation is well posed and study what the limit is when the noise level tends to zero.

Series: Probability Working Seminar

It is well known that isoperimetric type inequalities can imply concentration inequalities, but the reverse is not true generally. However, recently E Milman and M Ledoux proved that under some convex assumption of the Ricci curvature, the reverse is true in the Riemannian manifold setting. In this talk, we will focus on the semigroup tools in their papers. First, we introduce some classic methods to obtain concentration inequalities, i.e. from isoperimetric inequalities, Poincare's inequalities, log-Sobolev inequalities, and transportation inequalities. Second, by using semigroup tools, we will prove some kind of concentration inequalities, which then implies linear isoperimetry and super isoperimetry.

Series: Probability Working Seminar

In this talk I will present an elementary short proof of the existence of global in time H¨older continuous solutions for the Stochastic Navier-Stokes equation with small initial data ( in both, 3 and 2 dimensions). The proof is based on a Stochastic Lagrangian formulation of the Navier-Strokes equations. This talk summarizes several papers by Iyer, Mattingly and Constantin.

Series: Probability Working Seminar

This is a continuation of last week's talk.

Series: Probability Working Seminar

In my talk, I will present the main results of a recent article by Martin Hairer and Jonathan Mattingly on an ergodic theorem for Markov chains. I will consider Markov chains evolving in discrete time on an abstract, possibly uncountable, state space. Under certain regularity assumptions on the chain's transition kernel, such as the existence of a Foster-Lyapunov function with small level sets (what exactly is meant by that will be thoroughly explained in the talk), one can establish the existence and uniqueness of a stationary distribution. I will focus on a new proof technique for that theorem which relies on a family of metrics on the set of probability measures living on the state space. The main result of my talk will be a strict contraction estimate involving these metrics.

Series: Probability Working Seminar

The talk is based on a 1992 paper by Yakov Sinai. He proves a localization property for random walks in the random potential known as Nechaev's model.

Series: Probability Working Seminar

The talk is based on the paper by B. Klartag. It will be shown that there exists a sequence \eps_n\to 0 for which the
following holds: let K be a compact convex subset in R^n with nonempty
interior and X a random vector uniformly distributed in K. Then there
exists a unit vector v, a real number \theta and \sigma^2>0 such that
d_TV(, Z)\leq \eps_n
where Z has Normal(\theta,\sigma^2) distribution and d_TV - the total
variation distance. Under some additional assumptions on X, the
statement is true for most vectors v \in R^n.

Series: Probability Working Seminar

In this talk, we will introduce the classical Cramer's Theorem. The
pattern of proof is one of the two most powerful tools in the theory
of large deviations. Namely, the upper bound comes from optimizing
over a family of Chebychef inequalities; while the lower bound comes
from introducing a Radon-Dikodym factor in order to make what was
originally "deviant" behavior look like typical behavior.
If time permits, we will extend the Cramer's Theorem to a more general
setting and discuss the Sanov Theorem.
This talk is based on Deuschel and Stroock's .