Маленький семинар по теории вероятности

Fall 2018

In analysis, a typical inequality of the Hardy-Landau-Kolmogorov type connects, and usually upper bounds, norms of intermediate derivatives of sufficiently smooth functions on the real line or a segment of it by using norms of higher order derivatives. The inequalities usually come with universal constants in front, and in some cases, mostly because of Kolmogorov's work, the best constant is known, at least in principle.
What do these have to do with statistics? We will describe in this seminar how to combine these inequalities with other inequalities in other domains of analysis to ultimately produce bounds, explicit, on the total variation diameters of what statisticians call shape-restricted non-parametric densities. For example, the shape restriction could be monotonicity, or convexity, or logconcavity. The inequalities that the Hardy-Landau-Kolmorov inequalities are combined with are convexity inequalities. For example, Young's inequality, or Ostrowski's inequality, etc.
Are there any practical uses? Yes, powers of likelihood ratio tests, or, local limit theorems.

In its most well-known form, the Loewner equation gives a correspondence between curves in the upper half-plane and continuous real functions (called the 'driving function' for the equation). We consider the generalized Loewner equation, where the driving function has been replaced by a time-dependent real measure. In the first part of the talk, we investigate the delicate relationship between the driving measure and the generated hull, specifying a class of discrete random driving measures that generate hulls in the upper half-plane that are embeddings of trees. In the second part of the talk, we consider the probabilistic question of finding the scaling limit of these measures for Galton-Watons trees. We particularly focus on distributions on trees converging to the continuum random tree, and we conclude by describing connections to the complex Burgers equation. Joint work with Govind Menon (Brown University).

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\scriptsize R^{d}}$. We consider the case where the points of the Čech complex are generated by a Poisson process with intensity ${\scriptsize n_f}$ for a probability density ${\scriptsize f}$. We look at the cases where the behavior of the connectivity radius of Čech complex causes simplices of dimension greater than ${\scriptsize k + 1}$ to vanish in probability, the so-called sparse and Poisson regimes, as well when the connectivity radius is on the order of ${\scriptsize n−1/d}$, the critical regime. We establish limit theorems in all of the aforementioned regimes, a central limit theorem for the sparse and critical regimes, and a Poisson limit theorem for the Poisson regime. When the connectivity radius of the Čech complex is ${\scriptsize o(n−1/d)}$, i.e., the sparse and Poisson regimes, we can decompose the limiting processes into a time-changed Brownian motion and a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension. This is joint work with Takashi Owada (Purdue).

Coupling is a way of constructing Markov processes with prescribed laws on the same space. In 1991, M. Cranston was the first to show the connection between coupling and analytic gradient estimates. We will start by introducing the notion of coupling and review the known results for the elliptic setting. In this talk, we consider coupling of two hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). We construct a successful non-Markovian coupling on the Heisenberg group, which is the first example of a nontrivial sub-Riemannian manifold. We use this coupling to obtain gradient estimates similar to the one’s Cranston obtained on Riemannian manifolds. We will also obtain similar results for the degenerate Kolmogorov diffusion. This talk is based on joint work with Sayan Banerjee and Maria Gordina.

In this talk we study small time asymptotic of the heat content for a smoothly bounded domain with non-characteristic boundary in the Heisenberg group, which captures geometric information of the of the boundary including perimeter and the total horizontal mean curvature of the boundary of the domain. We use probabilistic method by studying the escaping probability of the horizon- tal Brownian motion process that is canonically associated to the sub-Riemannian structure of the Heisenberg group. This is a joint work with J. Tyson.

We compute the limit of the free energy of the mean field generated by the independent Brownian particles pairwise interacting through a non-negative definite function. Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system. This talk is based on a joint work with Tuoc V. Phan.

Let ${\scriptsize T}$ be a tree chosen from Galton-Watson measure and let ${\scriptsize \{U_v\}}$ be IID uniform ${\scriptsize [0,1]}$ random variables associated with the edges between each vertex and its parent. These define coupled Bernoulli percolation processes, as well as an invasion percolation process. We study quenched properties of these percolations: properties conditional on ${\scriptsize T}$ that hold for almost every ${\scriptsize T}$. The invasion cluster has a backbone decomposition which is Markovian if you put on the right blinders. Under suitable moment conditions, the law of the (a.s. unique) backbone ray is absolutely continuous with respect to limit uniform measure. The quenched survival probabilities are smooth in the supercritical region ${\scriptsize p > p_c}$. Their behavior as ${\scriptsize p \to p_c}$ depends on moment assumptions for the offspring distribution

In many situations in mechanics movement is constrained to `horizontal directions'. In the simplest settings this corresponds to moving along submanifolds; the opposite extreme of manifolds with a `completely non-integrable' collection of horizontal directions are known as contact manifolds and arise often, e.g., as level sets of Hamiltonians. Questions in analysis and geometry have natural analogues on contact manifolds but with differentiation and dynamics restricted to horizontal directions. I will report on joint work with P. Albin and J. Wang in which these spaces are studied as limits of Riemannian manifolds where movement in the non-horizontal directions is heavily penalized. The `horizontal Laplacian’ (the infinitesimal generator of horizontal Brownian motion) is shown to be the limit of the Riemannian Laplacians in the strong sense that the heat kernels converge smoothly, in an appropriate sense. If time permits I will discuss the corresponding result for the Hodge Laplacians on differential forms which is complicated by the fact that the Laplacians no longer converge.

Some stochastic processes, such as recurrent Markov chains, posess a regenerative structure which gives an embedded i.i.d. structure inside the stochastic process. This i.i.d. structure can then be exploited to prove limiting results like laws of large numbers or central limit theorems. We show that the Berry-Esseen estimates on the error in the central limit theorems can also be extended to these regenerative processes. This had previously been done by Bolthausen under a third moment assumption, but we generalize the results to also cover cases with weaker moment assumptions. As applications of our main results we give quantitative CLT estimates for (1) additive functionals of Markov chains with sufficiently strong mixing conditions and (2) multi-dimensional random walks in random environments with non-zero limiting speed. This talk is based on joint work with Xiaoqin Guo.

We prove a discrete Beurling estimate for the harmonic measure in a wedge in ${\scriptsize \mathbf{Z}^2}$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than ${\scriptsize \pi/4}$ stabilizes. This allows to consider the infinite DLA as a finite time growth process and questions about the number of arms, growth and dimension. I will present some conjectures and open problems. This is joint work with Ron Rosenthal (Technion) and Yuan Zhang (Pekin University).

We study the effective behavior of a Brownian motion in both one and two dimensional comb like domains. This problem arises in a variety of physical situations such as transport in tissues, and linear porous media. We show convergence to a limiting process when when both the spacing between the teeth, and the probability of entering a tooth vanish at the same rate. This limiting process exhibits an anomalous diffusive behavior, and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. At the PDE level, this leads to equations that have fractional time derivatives and are similar to the Bassett differential equation.

We consider the parabolic Anderson model in one spatial dimension driven by a time-independent Gaussian noise ${\scriptsize W}$, which has the covariance structure of a fractional Brownian motion with Hurst parameter ${\scriptsize H}$. We consider the case ${\scriptsize H < \frac{1}{2}}$ and establish existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form ${\scriptsize \frac{1}{2}\Delta + \dot{W}}$.

A function's excursion set is the sub-domain where its value exceeds some threshold. Some key examples illustrating the central role that excursion sets play in different application areas are as follows. In medical imaging, in order to understand the brain parts involved in a particular task, analysts frequently look at the high blood flow level regions in the brain when the said task is being performed. In control theory, it is known that the viability and invariance properties of control systems can be expressed as super-level sets of suitable value functions. In robotics, in order to plan its motion, a sensor robot may want to identify the sub-terrain where the accessibility probability is above some threshold. Often, functions whose excursions are of interest are either random themselves (for e.g., due to noise) or, while being deterministic, are too complicated and hence can be treated as being a sample of a random field. In this sense, studying the topology of random field excursions is vital. This work is the first detailed study of their Betti numbers (number of holes) in the so-called `sparse' regime. Specifically, we consider a piecewise constant Gaussian field whose covariance function is positive and satisfies some local, boundedness, and decay rate conditions. We model its excursion set via a Cech complex. For Betti numbers of this complex, we then prove various limit theorems as the window size and the excursion level together grow to infinity. Our results include asymptotic mean and variance estimates, a vanishing to non-vanishing phase transition with a precise estimate of the transition threshold, and a weak law in the non-vanishing regime. We further have a Poisson approximation and a central limit theorem close to the transition threshold. Our proofs combine extreme value theory and combinatorial topology tools. This is joint work with Sunder Ram Krishnan.