Tuesday, Oct 7
2008 : Clark Hall 700
|09:30-10:30||Anja Sturm: Interface tightness for long-range voter models with
|11:00-12:00||Elizabeth Meckes: When is normal normal?|
Tai Melcher: Hypoelliptic diffusions and potential theory|
There is a long standing and deep relationship between potential theory and probability.
In the particular case of Brownian motion on a Riemannian manifold, much of the behavior of this process can be understood via equivalent potential theoretic statements and vice versa. We will discuss some aspects of this theory in the context of hypoelliptic
diffusions, the natural analogue of Brownian motion on certain manifolds.
Masha Gordina: Gaussian type measures and Riemannian geometry in
We will talk about how curvature of an infinite-dimensional curved space effects the
behaviour of Gaussian type measures. In particular, several settings for
infinite-dimensional manifolds will be considered: Hilbert-Schmidt groups which are
natural infinite-dimensional analogues of matrix groups, Heisenberg infinite-dimensional
groups modelled over an abstract Wiener space, and the homogeneous space Diff(S^1)/S^1 associated with the Virasoro algebra. We will describe what is known about the Ricci
curvature in each of the case, and how its boundness (or unboundness) is reflected in the heat kernel (Gaussian) measure behaviour. The work on the Heisenberg group is joint with Bruce Driver
Nina Gantert: Biased random walk on percolation clusters (of trees)|
The following model is considered in the physics literature as a model for transport in an inhomogeneous medium. Let p>1/2 and perform i.i.d. bond percolation on Z^2. Consider a random walk on the (unique) infinite cluster which has a bias to the right. It has been shown that, for all values of 1/2< p <1, the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive. This proves part of the predictions made in the physics literature. We describe some of the (many) open questions on the model. We then address the convergence in law for the simpler model of biased random walks on percolation clusters of trees. We consider a supercritical Galton-Watson tree, conditioned on survival (in particular, this includes the case of the infinite percolation cluster of a regular tree) and run a biased random walk on this tree. We investigate the distributions of the walker at time n and show that they are tight, but do in general not converge. The talk is based on joint works with Noam Berger, Gerard Ben Arous, Alexander Fribergh, Alan Hammond and Yuval Peres.
Elena Kosygina: Positively and negatively excited random walks on integers|
We consider excited random walks on integers with a bounded number of
i.i.d. cookies per site which may induce drifts both to the left and
to the right. We extend the criteria for recurrence and transience by
M. Zerner and for positivity of speed by A.-L. Basdevant and
A. Singh to this case and also prove an annealed central limit
theorem. The proofs are based on results from the literature
concerning branching processes with migration and make use of a
certain renewal structure.
Kavita Ramanan: Properties of Reflected Diffusions |
Reflected diffusions arise in a variety of contexts, including
as approximations to stochastic networks, in economics and in the study of
random matrices. We address several questions related to fundamental
properties of these diffusions, which are also of interest in
applications. The answers reveal some interesting connections with convex
analysis and algebra. The talk is based on various joint works with
Rami Atar, Amarjit Budhiraja, Chris Burdzy, Paul Dupuis, Weining Kang and
Ruth Williams: Stochastic networks with resource sharing|
Stochastic networks are used as models for complex systems involving dynamic interactions subject to uncertainty. Application domains include manufacturing, the service industry, telecommunications, and computer systems. Networks arising in modern applications are often highly complex and heterogeneous, with network features that transcend those of conventional queueing models. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application
will be used to illustrate a general approach to the study of stochastic networks
using more tractable approximate models. Specifically, we consider a
data network model that represents the randomly varying number of flows present in a network where bandwidth is shared fairly amongst elastic documents. This model, introduced by Massoulie and Roberts, can be viewed as a stochastic network with simultaneous resource possession. Elegant fluid and diffusion approximations will be used to study the performance of this model.
The talk will conclude with a summary of the current status and description of open problems associated with the further development of approximate models for general stochastic networks. This talk is based in part on joint work with W. N. Kang,
F. P. Kelly, and N. H. Lee.
Deena Schmidt: Waiting for two mutations with applications to DNA regulatory sequence evolution and the limits of Darwinian evolution|
Results of Nowak and collaborators concerning the onset of cancer due to the
inactivation of tumor suppressor genes give the distribution of time until
some individual in a population has experienced two prespecified mutations,
and the time until this mutant phenotype becomes fixed in the population. We
apply and extend these results to obtain insights into DNA regulatory
sequence evolution in Drosophila (fruit flies) and humans. In particular, we
examine the waiting time for a pair of mutations, the first of which
inactivates an existing transcription factor binding site and the second
which creates a new one. Consistent with recent experimental
observations for Drosophila, we find that a few million years is sufficient,
but for humans with a much smaller effective population size, this type of
change would take more than 100 million years. In addition, we use these
results to expose flaws in some of Michael Behe's arguments concerning
mathematical limits to Darwinian evolution.
Anita Winter: Geometry of spaces of trees and the tree-valued diffusion arising from the 'Aldous move on cladograms'|
Metric probability trees are complete and separable 0-hyperbolic metric spaces
equipped with a probability measure. We say that a sequence of metric probability trees are converging in the Gromov-weak topology to a limit metric probability tree if all
subtrees spanned by a finite sample converge.
The space of all metric probability trees equipped with the Gromov-weak topology serves as a state space for the limit dynamics of tree-valued Markov chains as the number of vertices increases.
In this talk we want to use techniques from martingale problems to contruct the candidate of the diffusion limit of the Aldous move on cladograms introduced by David Aldous in 1990. (joint work with Leonid Mytnik)
Anja Sturm: Interface tightness for long-range voter models with exclusion dynamics|
We prove a theorem on extinction versus unbounded growth for parity preserving
cancellative interacting particle systems. Here, extinction or growth refers to the long
term behavior of ones ("particles") if the system is started with finitely many ones
initially. The theorem is proved under the assumption that the one particle state is not
The result is then applied to parity preserving particle systems in one dimension, which
we can interpret as the interface dynamics of mixtures of long-range voter models and
exclusion process dynamics. We use this fact to show that the latter models exhibit
interface tightness, meaning that the trivial one interface state is positively
recurrent, provided that the infection rates have a finite second moment.
This is joint work with Jan Swart (UTIA Prague).
Elizabeth Meckes: When is normal normal?|
There is a large class of results which say that, for parametrized collection of
random variables, a random variable from the collection behaves a certain way for
``most'' values of the parameter. A nice example of such a result is a theorem of Persi
Diaconis and David Freedman, which roughly says that if you have a large collection of
high-dimensional data points, most one-dimensional projections of the data will look
Gaussian even if the data have no particular structure. In this talk, I'll discuss a
quantitative version of this result which aims to address the question, ``When is an
almost-Gaussian projection interesting?'' I'll give an overview of the proof, which uses
Stein's method, the concentration of measure phenomenon, and the geometry of certain