Fall 2012

We prove the boundedness on $L^p$, $1 < p < \infty$, of operators on manifolds which arise by taking conditional expectation of transformations of stochastic integrals. These operators include various classical operators such as second order Riesz transforms and operators of Laplace transform-type.

Joint work with Rodrigo Banuelos.

Gaussian dilations and convolutions are often used as models in probability and statistics, and also arise on their own in the course of theoretical analysis of many problems. All symmetric stable laws are dilations, but none, other than the Gaussian itself, can be a convolution. Unimodal distributions obtained by an infinitely divisible mixing of uniforms cannot be in the convolution class. The convolution class without dilation is rather narrow, but measures external to it have very close projections: the class has nice tentacles.

Convolutions increase uncertainty. So, the tails of convolutions would be appropriately heavier than the component tails. Convolution tails have arisen as key quantifiers in prior work of various authors, notably in biostatistics. We recently had it emerge in thresholding estimation of Gaussian means when we studied Bayes risks of thresholding estimates, DasGupta and Johnstone, 2012. Three things play together to determine the asymptotic behavior of the Bayes risk; the prior spread, the thresholding constant, and an old friend from real analysis, the theory of regular variation.

Excited random walks (also called cookie random walks) are self-interacting random walks where the transition probabilities depend on the number of previous visits to the current location. Although the models are quite different, many of the known results for one-dimensional excited random walks have turned out to be remarkably similar to the corresponding results for random walks in random environments. For instance, one can have transience with sub-linear speed and limiting distributions that are non-Gaussian. In this talk I will prove a large deviation principle for excited random walks.

The main tool used will be what is known as the "backwards branching process" associated with the excited random walk, thus reducing the problem to proving a large deviation principle for the empirical mean of a Markov chain (a much simpler task). While we do not obtain an explicit formula for the large deviation rate function, we will be able to give a good qualitative description of the rate function. While many features of the rate function are similar to the corresponding rate function for RWRE, there are some interesting differences that highlight the major difference between RWRE and excited random walks.

We study the stochastic behavior of parameters of a network using a tree model of growth, from building blocks that are themselves rooted trees. We give asymptotic results about the number of leaves, depth of nodes, total path length and height of such trees.

Joint work with Hosam Mahmoud (George Washington University) and Mark Daniel Ward (Purdue).

The "hard-core" distribution on a graph $G$ is the probability distribution on the independent sets of $G$ (sets of mutually non-adjacent vertices) in which each such set $I$ has probability proportional to $\lambda^{|I|}$, for some $\lambda > 0$.

The hard-core distribution arose as a simple model of the occupation of space by a gas with massive particles, and is mainly of interest because it has the potential to exhibit a liquid-solid phase transition: for small $\lambda$ a typical configuration should be a mostly uncorrelated sparse set of vertices, while for larger $\lambda$ it should be a highly correlated dense subset of a maximum independent set.

I'll focus on the integer lattice ${\mathbb Z}^2$, where we strongly expect a liquid-solid transition point to exist. I'll discuss recent work with Blanca, Randall and Tetali, where we show that the solid phase can be better understood by introducing a new class of self-avoiding walks on ${\mathbb Z}^2$ that mimics the movement of taxi cabs around Manhattan.

We study the sub-Laplacian $L$ on a compact Riemannian manifold $\mathbb{M}$. $L$ is a symmetric diffusion operator which is subelliptic but nowhere elliptic. We obtain the Bakry-Émery type criterion (curvature-dimension inequality) for $L$ which gives an analytic approach to the geometric property. As a consequence, we prove that under suitable geometric bounds, spectral gap estimates can be obtained as well as the convergence to the equilibrium of the associated Markov processes.

We present a family of non-linear noisy heat equations that have "intermittent" and/or "chaotic" behavior. Among other things, we shall see that a characteristic feature of many such noisy PDEs is that they develop "shocks." All terms in quotations will be made precise during the talk.

This is based on joint work with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu.

In this talk, I'll present extensions of results of M. van den Berg on two-term asymptotics for the trace of Schodinger operators when the Laplacian, the generator of the Brownian motion, is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and other closely related Levy processes. (joint work with R. Banuelos)

We provide a joint estimate for the drift and Hurst parameters of a stationary Ornstein-Uhlenbeck (fOU) process driven by a fractional Brownian motion. This estimator is based on a system of moment equations involving filters of different orders and the autocovariance function of the fOU process. This idea can be generalized to a certain class of stationary Gaussian processes with explicit spectral density. Using the asymptotic theory of the Generalized Method of Moments, the consistency and asymptotic normality of this estimator are established. Finally, a simulation study is done under different scenarios. (Joint work with Prof. Frederi Viens.)

Probabilistically, the ``arbitrage-free price" of at-the-money (ATM) European options can be expressed as the expected value of the functional $(\exp(X_{t})-1)_+$, where $X_{t}$ is the so-called log-return process $\ln(S_{t}/S_{0})$ of the underlying stock price process $S_{t}$. These instruments are the most tradable options in the market and they are often used to ``calibrate" the model's parameters and hedging other more exotic derivatives. In this talk, we obtain the ``second-order" asymptotic behavior of ATM option prices as the maturity $t$ goes to 0 for a tempered stable Lévy processes $X$. Tempered stable processes can be constructed from their stable counterparts via a change of probability measure. Unlike the stable distributions, tempered stables may exhibit finite high-order moments while preserving the stable's self-similar behavior in short-time. Our method of proof is based on an integral representation of the option price involving the tail probability $P(X_{t}>{}x)$ and a suitable change of probability measure under which the process becomes stable. This is a joint work with R. Gong and C. Houdré.

Is there a natural way to put a random total ordering on the vertices of a finite graph? Natural here means that all finite graphs get an isomorphism-invariant random ordering and induced subgraphs get the random ordering that is inherited from the larger graph. Thus, the uniformly random ordering is natural; are there any others? What if we restrict to certain kinds of graphs? What about finite vector spaces over a finite field or finite metric spaces? We discuss these questions and sketch how their answers give unique ergodicity of corresponding automorphism groups; for example, in the case of graphs, the group is the automorphism group of the infinite random graph. This is joint work with Omer Angel and Alexander Kechris.

Let $A_n$ be an $n\times n$ matrix whose entries are independent random variables of sub-gaussian distribution with mean zero and variance one. In this talk we show that the determinant of $A_n$ follows a log-normal law. (joint with Van Vu)

In this talk we discuss new methods of quality issues for systems of stochastic Ito differential equations. These methods are associated with the construction for a stochastic system a special system of ordinary differential equations, in which the properties of the solutions can be judged on the properties of solutions of the original stochastic system. In this way, the following objectives are achieved:
1) Study of the limit ($L^2$ and almost sure) behavior of stochastic dynamic systems.
2) Study of the existence of deterministic invariantsets for stochastic systems.
3) An investigation of the stability of such sets in various probabilistic senses.
4) Development of Sturm oscillation solutions theory for linear stochastic equations of the second order.