| Weak quenched limiting distributions of a one-dimensional random walk in a random environment
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Jon Peterson (Cornell)
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.
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| Gaussian type analysis on infinite-dimensional Heisenberg groups
(pdf)
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Masha Gordina (U Connecticut)
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.
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| Hypoelliptic diffusions and heat kernels on Lie groups
(pdf)
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Nate Eldredge (Cornell)
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.
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| Steady-state simulation
of stochastic fluid networks with Lévy input (pdf) |
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Jose Blanchet (Columbia)
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.)
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| Large-scale measurement of
human behavior
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Sharad Goel (Yahoo Research)
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.
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| Probabilities of all real
zeros for random polynomials
(pdf)
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Wenbo Li (U Delaware)
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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