Probability Archives for Academic Year 2021

Dynamic polymers: invariant measures and ordering by noise

When: Thu, October 21, 2021 - 11:00am
Where: 0104
Speaker: Yuri Bakhtin (NYU) -
Abstract: Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.

A Decomposition-Separation Theorem for Markov Chains and Related Problems

When: Thu, October 28, 2021 - 11:00am
Where: Kirwan Hall 3206
Speaker: Isaac Sonin (UNC Charlotte) -
Abstract: Let $(P_{n}), n=0,1,2,...$ be a sequence of finite stochastic matrices of the same size $N\times N$. How does the product $\prod_{i=k}^nP_i, 0\leq k$, behave, when $n\rightarrow \infty$, if there are no assumptions about sequence the $(P_{n})$? In probabilistic terms this question is equivalent to a question about the asymptotic behaviour of a family of nonhomogeneous Markov chains defined by a sequence $(P_{n})$ and all possible initial distributions on a finite set $S, |S|=N$ with an initial time point $k$.

The surprising answer to this question is given by a fundamental theorem, which we call the Decomposition-Separation (DS)\ Theorem. This theorem was initiated by a small paper of A. N. Kolmogorov (1936) and formulated and proved in a few stages in a series of papers: D. Blackwell (1945), H. Cohn (1971,..., 1989) and I. Sonin (1987, 1991,..., 2008).

The DS Theorem has a simple deterministic interpretation in terms of the behaviour of the simplest model of an irreversible process. Such a model can be defined, for
example, by a finite number of cups filled with a solution (e.g. tea) having
possibly different concentrations and exchanging this solution at
discrete times. This can also be considered in terms of the evolution of a
systems of particles, which connects this theorem with a broad area called Consensus Algorithms.

Some new results will also be presented, but generally the DS
Theorem leaves many open problems and probably leads to generalizations in other fields of mathematics besides probability theory.

Among other topics that will be briefly mentioned are: The Markov Chain Tree Theorem and the question: "What does it mean that two events are independent?"

Rare exit events near a repelling equilibrium

When: Wed, November 3, 2021 - 2:00pm
Where: Kirwan Hall 3206
Speaker: Hong-Bin Chen (NYU) -
Abstract: Consider an ODE and a bounded domain in a Euclidean space. Assume, in that domain, the ODE has only one equilibrium and it is repelling. Under white noise perturbation, there is a positive probability that the dynamics emitting from the equilibrium point eventually exits the domain. We will discuss the exact asymptotics of rare exit events, e.g. exit locations and exit times, in the vanishing noise limit. It turns out that the decay rates of such rare events are polynomial powers of the noise magnitude, different from the exponential decay in the Freidlin-Wentzell large deviation theory. This talk is based on the joint work with Yuri Bakhtin.

KPZ on torus

When: Wed, November 17, 2021 - 2:00pm
Where: Physics 1410
Speaker: Yu Gu (UMD) -
Abstract: I will present a joint work with Tomasz Komorowski in which we establish the central limit theorem for the Kardar-Parisi-Zhang equation on a torus. I will explain the connection to the central limit theorem of the product of independent random matrices.