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UVA Probability Seminar
Mondays, 2:00 - 2:50pm
Kerchof 326
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Organizers: Christian Gromoll & Tai Melcher
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Fall 2010

27 Aug* Organizational meeting
30 Aug no seminar
6 Sep Gianluca Guadagni
What is a Random Walk in a Random Environment?
13 Sep Jon Peterson (Cornell)
Weak quenched limiting distributions of a one-dimensional random walk in a random environment
20 Sep Tai Melcher
What is Brownian motion on a (possibly infinite dimensional) Lie group?
27 Sep Masha Gordina (U Connecticut)
Gaussian type analysis on infinite-dimensional Heisenberg groups
4 Oct Nate Eldredge (Cornell)
Hypoelliptic diffusions and heat kernels on Lie groups
11 Oct no seminar: Fall Break
18 Oct Dan Dobbs
What is the Dichotomy Theorem?
25 Oct Marty Keutel
What is Reflected Brownian Motion?
1 Nov Jose Blanchet (Columbia)
Steady-state simulation of stochastic fluid networks with Lévy input
8 Nov no seminar
15 Nov Tipan Virella
What is Social Computing?
22 Nov Sharad Goel (Yahoo Research)
Large-scale measurement of human behavior
29 Nov Justin Webster
What are Random Polynomials?
6 Dec Wenbo Li (U Delaware)
Probabilities of all real zeros for random polynomials
* please note time, date, and/or location change

Abstracts

Weak quenched limiting distributions of a one-dimensional random walk in a random environment
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Jon Peterson (Cornell)
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In 1975 Kesten, Kozlov, and Spitzer computed the limiting distribution for a one-dimensional transient random walk in a random environment (RWRE) under the averaged measure. Somewhat surprisingly, the limiting distributions are not always Gaussian and are related to the stable distributions. In this talk I will consider the distribution of the RWRE under the quenched measure (i.e., conditioned on the environment). In previous work with Ofer Zeitouni I showed that for certain distributions on environments there does not exist an almost sure quenched limiting distribution. That is, for a fixed environment the distribution of the random walk (centered and scaled) does not converge to a deterministic distribution. This talk is based on ongoing research with Gennady Samorodnitsky in which we prove a weak quenched limiting distribution. That is, the quenched distribution (viewed as a random probability measure) converges in distribution on the space of random probability measures.

Gaussian type analysis on infinite-dimensional Heisenberg groups (pdf)
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Masha Gordina (U Connecticut)
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Infinite-dimensional Heisenberg groups and algebras come up in a number of applications motivated by physics, including Kac-Moody algebras. At the same time they proved a nice toy model for a number of questions in analysis over infinite-dimensional curved spaces. The Heisenberg groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the heat kernel measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter is a version of the Ito-Wiener expansion in the non-commutative setting. This is a joint work with B.Driver.

Hypoelliptic diffusions and heat kernels on Lie groups (pdf)
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Nate Eldredge (Cornell)
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Hypoelliptic diffusions are an interesting class of stochastic processes where, in spite of having a degenerate elliptic generator, the process is still able to wander throughout its state space. Many familiar facts about elliptic partial differential operators such as the Laplacian have analogues in the hypoelliptic setting. In this talk, I will discuss some of the basic ideas connected with hypoelliptic diffusions, including some notions of sub-Riemannian geometry, as well as some results regarding heat kernel estimates on a specific class of Lie groups.

Steady-state simulation of stochastic fluid networks with Lévy input (pdf)
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Jose Blanchet (Columbia)
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Reflected stochastic networks arise in the analysis of a large class of queueing systems. The most popular model of this type is perhaps reflected Brownian motion, which arises in the heavy-traffic analysis of generalized Jackson networks. In this talk we discuss Monte Carlo simulation strategies for the steady-state analysis of reflected stochastic networks. In particular, we show how to exactly simulate a reflected stochastic network with compound Poisson input and how to provide samples that are close (with explicit and controlled error bounds) to both the transient and the steady-state distribution of reflected Brownian motion in the positive orthant. (Joint work with Xinyun Chen.)

Large-scale measurement of human behavior (pdf)
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Sharad Goel (Yahoo Research)
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With the increasing availability of network and behavioral data---from what we buy, to where we travel, to whom we know---we are now able to observe and quantify social processes to a degree that would have seemed impossible just a decade ago. These new microscopes into human activity not only have substantive implications for economics, sociology, and psychology, but also raise challenging computational questions in large-scale data analysis. In this talk I'll present several illustrative examples from this emerging discipline of computational social science.


Probabilities of all real zeros for random polynomials (pdf)
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Wenbo Li (U Delaware)
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There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions.


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