Northwest Probability Seminar
The Seventh Northwest Probability Seminar
October 22, 2005
Lecture in Probability will be delivered by Charles Newman (Courant Institute,
New York University) in 2005.
Northwest Probability Seminars are one-day
mini-conferences held at the University of Washington
and organized in collaboration with
the Oregon State University, the University of British Columbia,
the University of Oregon, and the Theory Group at the Microsoft
Research. There is no registration fee. Participants
are requested to contact Zhen-Qing Chen
) in advance
so that adequate facilities may be arranged for.
The Scientific Committee for the NW Probability Seminar 2005
consists of Chris Burdzy (U Washington), Zhenqing Chen (U Washington),
Ed Perkins (U British Columbia), Qi-Man Shao (U Oregon) and Ed Waymire
(Oregon State U).
The talks will take place in Savery 239.
See the map
of north-central campus for the location of Savery Hall and
Padelford Hall (the Department of Mathematics is in the Padelford Hall).
campus maps are available at the UW Web site.
Parking on UW campus is free on Saturdays after 12:00 (noon).
More information is available at a
parking Web site
provided by UW.
This year, October 22, 2005 is also the Husky game day so the traffic
may be slow before and after the game.
- 10:30 Coffee and Registration - Savery Hall 241
- 11:00 Rami Atar
University of Technion and University of Washington
On Constrained Singular Control of Diffusions and Related PDE
A constrained singular control problem consists of minimizing
a cost associated with a process of the form
$W$ being Brownian motion, over processes $U$ that have
increments in a given cone and keep $X$ in the closure
of a given domain of $R^d$ for all times. Such problems arise,
in particular, in the study of stochastic queueing networks
in heavy traffic, and beyond dimension 1 they can rarely
be solved explicitly. PDE that characterize the value function
(i.e. Hamilton-Jacobi-Bellman equations with `state constraint'
boundary conditions) may be useful when explicit solutions are
not in hand. However, in two cases of the problem, that arise
in the applications mentioned above, standard techniques fail
to cover uniqueness and solvability for such PDE:
(a) unbounded domain and unbounded cost,
(b) bounded domain and cost that involves an improper integral
(with a function of $X$ as an integrand).
Combining probabilistic and PDE tools we establish unique
solvability for (a) and (b) above under appropriate conditions.
Some of the results are new even for $d=1$.
This is joint work with Amarjit Budhiraja and Ruth Williams.
- 12:00 Nathanael Berestycki
University of British Columbia
Small-time Behavior of Beta-coalescents
Lambda-coalescents were introduced by Pitman in (1999) and Sagitov
(1999). These processes describe the evolution of particles that
undergo stochastic coagulation in such a way that several blocks can
merge at the same time to form a single block. In the case where the
measure Lambda has the Beta(2-a,a) distribution, Birkner et al.
recently used the Donnelly-Kurtz lookdown construction to prove that
Beta-coalescents can be obtained as the time-changed genealogies of a
continuous-state branching process with stable branching mechanism.
Here we use this result to prove that Beta-coalescents can be further
embedded in continuous stable random trees, for which much is known
due to recent progress of Duquesne and Le Gall. This produces a
number of results concerning the small-time behavior of
Beta-coalescents. Most notably, we get an almost sure limit theorem
for the number of blocks at small times, for the rescaled sizes of
the blocks, and give the multifractal spectrum corresponding to the
emergence of blocks with atypical size. Also, we are able to find
asymptotics for several quantities of interest to biologists in the
context of population genetics.
This is joint work with Julien Berestycki (Univ. Marseille) and Jason
- 1:00 - 2:30 Lunch - Savery 241
- 2:30 Yevgeniy Kovchegov
Oregon State University
Generalized Symmetric Exclusion Processes
We will consider the particle systems
that interact via permutations, where transition rates are assigned not
to the jumps from a site to a site, but to the permutations themselves.
This is a way to generalize symmetric exclusion processes
that was suggested by T. Liggett. Recall that in case of symmetric
exclusion, particles interact via transpositions.
We develop a number of new couplings for these permutation
processes and establish the needed conditions for them to apply. We use
duality, couplings and other tools to explore the stationary distributions
for permutation processes with translation invariant rates.
- 3:30-3:45 Coffee - Savery Hall 241
- 3:45 Charles Newman
Courant Institute, New York University
"Birnbaum Lecture": Scaling Limit of two-dimensional critical percolation
We review the continuum nonsimple loop process
that represents the scaling limit of 2D critical percolation
and then, if time permits, discuss some ideas and open problems
associated with its extension to scaling limits of "near-critical"
percolation and minimal spanning trees.
- 5:30 No host dinner at
Cedars Restaurant on Brooklyn.
Click on the restaurant name to go to its Web page.