# Northwest Probability Seminar

### and Pacific Institute for Mathematical Sciences

The Birnbaum 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 (zchen@math.washington.edu ) 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). More 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.

### Schedule

• 10:30 Coffee and Registration - Savery Hall 241
• 11:00 Rami Atar (photo), 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
$$X=x+\int_0^.b(X)dt+\int_0^.\sigma(X)dW+\int_[0,.]g(X(.-))dU,$$
$W$ 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 (photo), 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 Schweinsberg (UCSD).

• 1:00 - 2:30 Lunch - Savery 241

• 2:30 Yevgeniy Kovchegov (photo), 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 (photo), 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.