Fall 2017

The existence and Hausdorff dimension of multiple points of paths have been intensely studied for Brownian motion and more general Lévy processes. The purpose of this talk is to present some new results in this area for a class of Lévy processes called operator stable Lévy processes. Under additional symmetry assumption, we will give an explicit dimension formula for the set of multiple points in terms of the eigenvalues of the stability exponent, which is a nonsingular square matrix. We will also discuss an existence criterion for multiple points. The talk is based on a joint project with Yimin Xiao.

The term limit theorem is associated with a multitude of statements having to do with the convergence of probability distributions of sums of increasing number of random variables. Given that a limit theorem result holds, weighted limit theorem in this talk is referred to the problem: what is the asymptotic behavior of the corresponding weighted sums? The problem has drawn a lot of attention in recent articles due to its essential role in topics such as time-discrete numerical schemes, parameter estimations, Ito’s formula in law, and normal approximations, and various unexpected weighted limit theorems have been discovered since then. The purpose of this talk is to introduce a universal and surprisingly simple framework for this problem, and to provide generalizations of the existing results in a few aspects.

The study of the stability of various classical inequalities in analysis and geometry (Sobolev, Hardy-Littlewood-Sobolev, Nash, Housdorff-Young, isoperimetric, Faber-Krahn, etc.) has received a lot of attention in recent years. Following the seminal work of Burkholder in the 80's, there has been considerable interest in sharp inequalities for martingales and their applications to various problems in analysis. Unlike many sharp inequalities in analysis and geometry, extremals for these inequalities do not exist and equality is never attained except for trivial cases. In this talk we discuss the nature and stability of the almost extremals in Burkholder's inequalities for differentially subordinate martingales and extensions to orthogonal martingales that arise in several problems. As an application we present similar results for the real and imaginary parts of the Beruling-Ahlfors operator and other singular integrals. This talk is based on join work with Adam Oscekowski of the University of Warsaw.

The logarithmic Sobolev inequality has been extensively studied in analysis and probability. After the equality case is fully characterized by E. Carlen, it is natural to ask how far an admissible function that is close to achieving the equality deviates from the optimizers. This question is called a stability for log Sobolev inequality. In this talk, we discuss stability results for Logarithmic Sobolev inequality in terms of the Wasserstein distance and the total variation distance. This is based on a joint work with Emanuel Indrei.

We are interested in a general class of recursive discrete dynamics : ${\scriptsize X_{n+1}=F(X_n,\Delta_{n+1})}$ where ${\scriptsize (\Delta_n)_{n\in \mathbb{Z}}}$ is an ergodic stationary Gaussian sequence. We can think for instance of fractional Brownian motion increments. First, we will see how it is possible to define invariant distributions in this a priori non-Markovian setting. Then, after proving existence of such a distribution, we will get a uniqueness result and a rate of convergence to this invariant distribution in total variation distance. The proof is based on a coupling method (with a step which is specific to this non-Markovian framework) first implemented by M.Hairer in a continuous time setting. This method was also used by J. Fontbona and F. Panloup, as well as A. Deya, F. Panloup and S. Tindel in order to extend M. Hairer's results.

In this talk we present some results that extends the Hormander condition to delayed diffusions. We will state a sufficient condition for the existence of density for the marginals of a delayed diffusion. This condition takes into account and quantifies the additional noise introduced to the system through the delay. The talk is based on joint work with Reda Chhaibi.

At each vertex of a graph ${\scriptsize G}$, independently place a car with probability ${\scriptsize p}$ and place a parking spot otherwise. Each car performs an independent random walk until it finds a vacant parking spot, at which time it parks in the spot. Each parking spot can be occupied by at most one car, with ties broken uniformly, and cars drive over occupied parking spots. Given ${\scriptsize G}$ and ${\scriptsize p}$, does every car eventually park? Do there exist parking spots that remain vacant forever? A phase transition in this model was recently observed by Lackner and Panholzer for uniform random directed trees on ${\scriptsize n}$ vertices, and by Goldschmidt and Przykucki on directed critical Galton-Watson trees. We prove that a phase transition occurs at ${\scriptsize p=1/2}$ when ${\scriptsize G}$ is any infinite, transitive, unimodular graph (including ${\scriptsize Z^d}$ and the ${\scriptsize (d+1)}$-regular tree). This is joint work with Michael Damron, Janko Gravner, Matthew Junge and Hanbaek Lyu.

The talk addresses the following general question: consider random power series or random infinite products, and ask what are their limiting distributions, with centering and norming, or the exact distribution, when they converge almost surely. Of particular importance in this seminar will be deleted series, or deleted products. For example, take the series for ${\scriptsize e^x}$, and sum over a subsequence of the natural numbers; or, take ${\scriptsize n!}$, and take the product over a random subset of ${\scriptsize\{1, 2, \ldots, n\}}$. The deleted series and the products are sometimes asymptotically Gaussian, and sometimes something else. The centering sequence depends on how sparse is the subsequence over which the deleted summation is performed. A famous case will be revisited: the Erdös-Kahane problem of the distribution of INFINITE BERNOULLI CONVOLUTIONS. The question that has attracted the most attention is when are infinite Bernoulli convolutions absolutely continuous. We know when they are not (those that arise from reciprocals of PV numbers; Erdös already knew it); we know a few explicit cases when they are, due to A. Garsia. The deepest but nonconcrete (to a practical person) are the results on almost sure absolute continuity of such convolutions, and (Hausdorff) dimensions of the complementary set. The seminar will pose a (seemingly) nice related (and unattacked?) problem.

In this talk we will discuss wetting models in (1+1) dimensions pinned to a shrinking strip. Except for the space geometry effect (the one dimensional lattice, in our case) these polymer models enjoy an interplay between two forces -- local (pinning) and global (entropic repulsion) -- and one expects a localization-delocalization phase transition to hold. When the strip size is fixed and the pinning function is constant and homogeneous, phase transition results are known, and moreover, the standard case (zero strip size) is completely solved and exhibits a sharp phase transition. In particular, the system behavior is drastically different in the three phases, sub-criticality, super criticality and criticality. We will focus on shrinking strips at criticality. Joint work in progress with Jean-Dominique Deuschel.

I will discuss mathematical aspects of my recent work on two related problems at the intersection of random graphs and information theory: (i) node arrival order inference -- for a dynamic random graph model, determine the extent to which the order in which nodes arrived can be inferred from the graph structure, and (ii) compression of structures -- for a given graph model, exhibit an efficiently computable and invertible mapping from unlabeled graphs to bit strings with minimum possible expected code length. Both problems are connected to the study of the symmetries of the graph model, as well as another combinatorial quantity -- the typical number of feasible labeled representatives of a given structure. I will focus on the case of the preferential attachment model, for which we are able to give a (nearly) complete characterization of the behavior of the size of the automorphism group, as well as a provably asymptotically optimal algorithm for (ii). I will also discuss progress on efficiently computable optimal estimators for certain natural formulations of (i).

We are dealing with the study of the validity of a large deviation principle for some nonlinear PDEs, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale ${\scriptsize\epsilon}$ and ${\scriptsize\delta}$, respectively, with ${\scriptsize \epsilon}$ and ${\scriptsize\delta}$ small. Depending on the relationship between ${\scriptsize\epsilon}$ and ${\scriptsize\delta}$ we will prove the validity of the large deviation principle in different functional spaces. We will illustrate our method by considering the two-dimensional stochastic Navier-Stokes equation and a class of stochastic reaction-diffusion equations, defined in any space dimension, including the dynamical ${\scriptsize\Phi^{2n}_d}$ model.

Understanding how two networks differ, or quantifying the degree to which a single network departs from a given model, is a challenging question in modern mathematical statistics. Here we show how subgraph densities, which for large graphs play a role analogous to moments in the context of random variables, enable a natural means of nonparametric network comparison. Coupled with a partial order derived from a notion of subgraph scale, we then show how this leads to an automated, computationally scalable comparison algorithm with provable properties. Joint work with P.-A. Maugis and S. C. Olhede; preprint here.

The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE.