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Program
| 9:00-9:50 | Sunder Sethuraman |
| 9:50-10:20 | Chris Orum |
| Monday morning | 10:20-10:40 | Break |
| 10:40-11:10 | Florian Sobieszky |
| 11:10-12:00 | Stefan Heinz |
| 12:00-2:00 | Lunch |
| 2:00-2:30 | Vicky Yang |
| 2:30-3:20 | Frank Gao |
| 3:20-3:50 | Jesse Gilbert |
| Monday afternoon | 3:50-4:10 | Break |
| 4:10-4:40 | Robertas Gabrys |
| 4:40-5:30 | David Aldous |
| 9:00-9:50 | Davar Khosnevisan |
| 9:50-10:20 | Burt Simon |
| Tuesday morning | 10:20-10:40 | Break |
| 10:40-11:10 | Manuel Lladser |
| 11:10-12:00 | Sergei Kuznetsov |
| 12:00-2:30 | Lunch |
| Tuesday
afternoon | 2:30-3:00 | Anatolii Puhalskii |
| 3:00-3:50 | Noam Berger |
Titles and Abstracts
David Aldous. Probability and Spatial
Networks.
Abstract. Network design and
analysis have been studied in many different applied contexts, yet
many simple-to-state abstracted mathematical problems have not been
studied very systematically. For a road network on n cities, what is
the trade-off between total network length and the efficiency of the
network in providing short routes? For an airline network on n cities,
requiring routes to have an average of no more that 3 hops, how short
can network length be? Such questions can involve probability in
several ways. First, the ``average case'' model of
randomly-distributed cities is a natural counterpart to worst-case
analysis. Second, while upper bounds on performance are obtained by
explicit construction, lower bounds need more mathematical arguments
provided by classical integral geometry. Third, the Poisson line
process turns out to be very useful!
(Joint work with Wilf Kendall. The two papers discussed are available
at
http://arxiv.org/abs/cond-mat/0702502 and
http://front.math.ucdavis.edu/math.PR/0701140)
Noam Berger. Detecting the trail of a random walker in a random
scenery.
Abstract. Flip a fair coin on each vertex of a transient graph. We call
this i.i.d. measure P. Then a random walker retosses the coins along its
path, this time with a bias. We call this new measure Q. Seeing the
configuration, can we tell whether it is a sample of P or of Q? In other
words, are P andd Q absolutely continuous or singular w.r.t. each other?
In this talk we answer this question for a large variety of graphs and
walks.
Robertas Gabrys. Portmanteau test of independence for functional
observations.
Abstract. In a number of fields, most notably finance and physical sciences, the time
series of finely spaced measurements form curves over some natural time inter-
val, e.g. a day or a week. Recent years have seen the development of tools for
analyzing such data which rely on concepts of Functional Data Analysis. To
validate the assumptions underlying these tools, it is important to apply some
test of independence to functional model errors or to suitably transformed
functional observations. We propose a χ2-test for independence and identi-
cal distribution which extends to the functional framework a well-established
univariate test. The test is easy to implement using the R package fda and
relies on the now standard functional principal component decomposition. It
has good empirical size and power which, in our simulations and examples,
is not affected by the choice of the functional basis. Its application is illus-
trated on two data sets: credit card sales activity and geomagnetic records.
Asymptotic theory based on correlations of matrix valued random variables,
functional principal component expansions and Hilbert space techniques is
developed.
Frank Gao. Metric Entropy Estimate of some shape-constrained function classes and
its small ball connection
Abstract. Shape constrained functions appear very commonly in nonparametric
estimation in statistics via renewal theory and mixing of uniform
distributions. Metric entropy estimate of these function classes is
needed because, as is well known, it determines the rate of
convergence of the nonparametric estimators such as the Maximum
Likelihood Estimator. In this talk, I will present some recent results
on entropy estimate of several shape-constrained multivariate function
classes, and on the small ball rate of their associated random
processes.
Jesse Gilbert. Tree Packings.
Abstract. We prove a variant of the Ringel-Kotzig conjecture. That is, we show
for all connected graphs H of size n, the complete graph K2n+1 has a
H-decomposition. We also pose several related questions.
Stefan Heinz. The Probabilistic Approach to Turbulence.
Abstract. The application of stochastic methods to turbulent flow simulations has significant advantages
compared to the use of deterministic methods: several important effects (chemical reactions)
can be treated exactly, and the closure problems of deterministic equations can be solved on
the basis of consistent models for the dynamics of turbulent fluctuations. However, stochastic
methods developed previously are still faced with a variety of problems. The talk describes
these problems and presents new solution strategies. The talk is organized in four parts. The
first part addresses the question of why the development of stochastic methods for turbulence
is needed. The second part describes the basics of stochastic methods for turbulent flows. The
modeling of molecular dynamics and turbulent velocity and scalar fields will be discussed.
Emphasis is placed on the explanation of problems and novel approaches for their solution.
The third part of the talk describes the application of stochastic methods to simulations of
turbulence. Both non-reacting and reacting turbulent flows will be considered. The fourth part
summarizes these developments and describes future activities.
Davar Khoshnevisan. Dynamical Processes.
Abstract. Consider a sequence of
i.i.d.\ random variables $X_1,X_2,\ldots\,.$ To each variable we
associated a rate-one Poisson clock, all independent of one another
and the $X_i$'s. When a clock ``rings,'' we replace the corresponding
$X_i$ with an i.i.d.\ copy. Let $X_i(t)$ denote the value of the $X_i$
variable at time $t$. Then $t\mapsto (X_1(t),X_2(t),\ldots)$ is a
strong Markov process whose invariant measure is the law of the
original sequence, $X_1,X_2,\ldots$, of random variables. This model
of equilibrium dynamics was introduced in H\"aggstr\"om, Peres, and
Steif (1997, 1998) and Benjamini, H\"aggstr\"om, Peres, and Steif
(2003), who ascribe the model to P. Malliavin. A variant appears
earlier in the work of Rusokov (1995).
When the $X_i$'s take the values zero and one, Benjamini, H\"aggstr\"om,
Peres, and Steif (2003) showed that there can be times $t$ when the
process $(X_1(t),X_2(t),\ldots)$, of zeros and ones, can have unusually
long runs of ones (say). R\'ev\'esz (2005) has made a conjecture about the
length of runs of ones that have a predescribed number of ``impurities.''
Here we describe:
1. A resolution of R\'ev\'esz's conjecture, and mention some of the
consequences of the method of proof;
2. A solution to a problem of Benjamini et al (2003) on a connection
between their parity test and the potential theory of Riesz; and
3. Time permitting, describe analogous problems for dynamical
percolation on trees.
Much of this material (1 and 2) is based on joint work with David Levin
and Pedro M\'endez.
Sergei Kuznetsov. On the equivalence of
traces for solutions of non-linear PDE.
Abstract. There exist two approaches
to the problem of classification of positive solutions of semilinear
PDE in terms of their boundary traces. One of them is (partly)
probabilistic, suggested by Dynkin and the speaker (DK-trace or fine
trace). The other is purely analytic, suggested by Marcus and Veron
(recently suggested MV-trace or exact trace). Since the definitions of
the traces are absolutely different, the relation between them is not
clear. We prove that the traces are, in fact, equivalent up to
indistinguishiability.
Manuel Lladser. Minimal Markov chain
representation of patterns problems.
Abstract. The study of regular patterns in random strings relies strongly on the
Markov chain embedding technique. This technique consists in embedding a
random string into a Markov chain that is at the same time informative of the
pattern of interest. An important application of this technique is the assessment
of patterns in RNA or DNA sequences: the main heuristic in genomic searches is
that over- or under-represented patterns in the genome are of biological
significance. The talk will characterize the smallest state-space size Markov
chain required to specify the exact or asymptotic distribution of the count
statistic of a regular pattern in the context of non-stationary Markov sources.
The characterization of such a chain is important to analyze patterns, which a
priori require exponentially large state spaces e.g. due to a large Markov order.
Some on-going research in the same lines but in the context of non-Markovian
strings will be also addressed during the talk.
Chris Orum. Branching processes and
Navier-Stokes equations.
Abstract. The model for the
analysis of Navier-Stokes equations that was introduced by Le Jan and
Sznitman (PTRF, Vol 109, No.3, 1997) involved representing the
solution as the expected value of a multiplicative functional defined
on a stochastic branching process. Some results stemming from the
line of research started by their paper will be discussed.
Anatolii Puhalskii. The large deviation
principle for join the shortest queue.
Abstract. A large deviation
principle is established for a broad class of join-the-shortest-queue
models. The action functional is expressed in terms of solutions to
mathematical programming problems. The large deviation limit point is
identified as a weak solution to a system of idempotent
equations. Uniqueness of the weak solution is proved by establishing
trajectorial uniqueness.
Sunder Sethuraman. On nonequilibrium fluctuations of a tagged particle in zero-range
interacting systems.
Abstract. The 'zero-range' particle system, one of several introduced in the '70's as
models of certain physical phenomena, follows a collection of random walks on a
lattice which interact infinitesimally only with those particles already at
their various locations.
In this talk, we consider the asymptotics of a distinguished, or tagged particle
in this interacting particle system. In particular, we discuss a
'nonequilibrium' invariance principle, in one dimension when the transition
rates are unbiased, with respect to a diffusion whose coefficients depend on the
'hydrodynamic' density, and give some open questions.
Burt Simon. Performance Analysis of a Basestation
Queue in a Wireless Communication Network.
Abstract. A basestation in a
wireless network is a receiver/transmitter that communicates with the
wireless devices in its coverage region ("cell"). We imagine a plane
with basestations scattered about. Each basestation has a fixed
coverage region, e.g., its Voronoi cell. Active wireless devices
appear and disappear as a (spatial) birth-death process. While a
device is active, it transmits data to its basestation at a rate b_i
that depends on the state i of a transient, continuous-time Markov
chain, Z(t). When Z(t) expires, the device "dies". We conjecture the
form of a heavy traffic limit for the basestation queue, which turns
out to be a reflected Brownian motion whose drift and variance terms
can be expressed explicity in terms of the basic model parameters. As
the average number of active devices in a cell increases, a properly
scaled version of the vector Q(t) (the number of devices in each state
of the Markov chain) converges to a certain multi-dimentional
Ornstein-Uhlenbeck process which can be used to approximate Q(t). The
auto-covariance function for the OU process is the same as for Q(t),
and the OU process is easier to work with in some
applications. Wireless devices outside of a given cell cause
interference with the devices within the cell, and this directly
affects the dynamics of Z(t). If the wireless devices are scattered
around the plane as a (spatial) Poisson process, and a few other
technical conditions are satisfied, we can derive the Laplace
transform of the stationary distribution of the total interference
level at a fixed location.
Florian Sobieczky. Strong amenability of horocyclic products of
Galton Watson trees.
Abstract. Certain percolative partial graphs of the horocyclic prod-
uct of two homogeneous trees are considered. Removal of edges by
a Bernoulli bond-percolation process is carried out only on a subset
of the set of edges. By its construction, the connected component
containing a preassigned root is the horocyclic product of two random
trees, sampled from the augmented Galton Watson measure. It is
shown, that it is almost surely amenable, if the offspring-distributions
of the two Galton Watson trees have supports with non-empty inter-
section. Given the case in which the percolation results from the horo-
cyclic product of two trees realized as samples of two identical indepen-
dent augmented Galton Watson measures, there is strong amenability,
almost surely. For a subclass of these random product graphs, suf-
ficient closeness to unsymmetric horocyclic products guarantees an-
chored expansion. In this case, we show the existence of a phase-
transition between strong amenability and weak non-amenability.
Vicky Yang. Estimation for Non-negative Levy-driven Ornstein-Uhlenbeck
Processes.
Abstract. The Ornstein-Uhlenbeck
process (or stationary continuous-time autoregression of order 1,
i.e. CAR(1)) driven by non-decreasing Levy process has been used to
model stochastic volatility of the log prices of financial assets (see
e.g. Barndorff-Nielsen and Shephard (2001)). In this talk, I will
present a highly efficient method of estimation for the parameters of
a CAR(1), taking advantage of the non-negativity of the driving
process. I will also show how to reconstruct the background driving
Levy process from a continuously observed realization of the process
and use this result to estimate the increments of the Levy process
itself when closely-spaced observations are available. Then, I derive
the asymptotic distribution of the coefficient estimator for a
gamma-driven CAR(1) and illustrate the performance of the procedure
through a simulation study. Lastly, a real dataset is analyzed using
our estimation procedure.
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List of speakers:
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