September 15, 2011

Behaviour near the extinction time in self-similar fragmentation chains

Christina Goldschmidt, Oxford University

Suppose we have a collection of blocks, which gradually split apart as time goes on. Each block waits an exponential amount of time with parameter given by its size to some power alpha, independently of the other blocks. Every block then splits randomly, but according to the same distribution. In this talk, I will focus on the case where alpha is negative, which means that smaller blocks split faster than larger ones. This gives rise to the phenomenon of loss of mass, whereby the smaller blocks split faster and faster until they are reduced to ``dust''. Indeed, it turns out that the whole state is reduced to dust in a finite time, almost surely (we call this the extinction time). A natural question is then: how do the block sizes behave as the process approaches its extinction time? The answer turns out to involve a somewhat unusual ``spine'' decomposition for the fragmentation, and Markov renewal theory. This is joint work (in progress!) with Bénédicte Haas (Paris-Dauphine).

September 29, 2011

Aldous' move on cladograms in the diffusion limit

Anita Winter, Universität Duisburg-Essen

A n-phylogenetic tree is a semi-labeled, unrooted and binary tree with n leaves labeled {1, 2, . . . , n} and with n-2 unlabeled internal leaves and positive edge lengths representing the time spans between common ancestors. In biological systematics n-phylogenetic trees are used to represent the evolutionary relationship between n species. If one does focus only on the kinship (that is taking all edge length of unit length), a more precise term is cladogram. Aldous constructed a Markov chain on cladograms and gave bounds on their mixing time. On the other hand, Aldous also gave a notion of convergence of cladograms which shows that the uniform cladogram with N leaves and edge length re-scaled by a factor of 1/\sqrt{N} converges to the so-called Brownian continuum random tree (CRT) which is the tree below“ a standard Brownian excursion and can be thought of as the uniform“ tree. These two results suggest that if we re-scale edge lengths by a factor of 1/\sqrt{N} and speeding up time by a factor of N^{3/2} the Aldous move on cladograms converges in some sense to a continuous tree-valued diffusion. The main emphasis of the talk is to give precise statements towards that direction. (This is joint work with Leonid Mytnik, Technion Haifa)

October 13, 2011

A simple path to asymptotics for the frontier of a branching Brownian motion

Matthew Roberts, McGill University

•  4:30-5:30pm at: McGill in Burnside Hall, room 920
Bramson's 1978 result on the position of the maximal particle in a branching Brownian motion has inspired many related results and generalizations in recent years. We show how modern methods can be used to give a relatively simple proof of the original result.

 

October 20, 2011

A simplified proof of the relation between scaling exponents in first-passage percolation

John McSweeney, Concordia University

Bistability is a common feature of many biochemical systems; for example, a gene's expression state may go from being fully ON to fully OFF and back at seemingly random times. The classical approach to studying a macroscopic chemical reaction system is to set up and analyze a system of ODEs for the species involved. However, if some species are present in small numbers, a first-principles stochastic formulation may be necessary in order to predict the qualitative behaviour of the system. In this talk we discuss a simple one-dimensional model where in addition to chemical reactions, the species counts are also affected by a cellular division mechanism. Depending on the orders of magnitude of the reaction constants, we find regimes where the concentration of a chemical species alternates between two extreme states but in qualitatively different ways, and that the variability introduced by the cellular division process may take us from one regime to another.

 

October 27, 2011

Nadaraya-Watson estimator for stochastic processes driven by stable Levy motions

Hongwei Long, Florida Atlantic University

•  4:30-5:30pm at: McGill in Burnside Hall, room 920
We discuss the non-parametric Nadaraya-Watson (N-W) estimator of the drift function for ergodic stochastic processes driven by stable noises and observed at discrete instants. Under geometrical mixing condition, we derive consistency and rate of convergence of the N-W estimator of the drift function. Furthermore, we obtain a central limit theorem for stable stochastic integrals. The central limit theorem has its own interest and is the crucial tool for the proofs of the main results. A simulation study illustrates the finite sample properties of the N-W estimator.

 

November 3, 2011

A simplified proof of the relation between scaling exponents in first-passage percolation

Michael Damron, Princeton University

In first passage percolation, we place i.i.d. non-negative weights on the nearestneighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes called the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.

 

November 10, 2011

Cancelled

Shunlong Luo, Chinese Academy of Sciences

November 17, 2011

The Aldous conjecture on a killed branching random walk

Olivier Zindy, Paris 6

We consider a branching random walk on R with an killing barrier at zero: starting from a nonnegative x, particles reproduce and move independently according to a certain point process, but are killed when they touch the negative half-line. In both critical and subcritical cases, the population dies out almost surely. We give the exact tail distribution of the total progeny of the killed branching random walk, which solves an open problem of D. Aldous. This is a joint work with Elie Aïdékon and Yueyun Hu.

 

November 24, 2011

Strong law of large numbers of a class of super-diffusions

Yan-xia Ren, Peking University

First I will state known results on strong law of large numbers for branching diffusions and super-diffusions. Then I will talk a recent result on strong law of large numbers for super-diffusions. In the recent work with Rong-Li Liu and Renming Song, we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions X corresponding to the evolution equation \partial_t u(t) = Lu(t) + \beta u(t) - \psi (u(t)) on a bounded domain D in R^d , where the branching mechanism \psi (x, \lambda ) = a\lambda ^2 + \int (e-\lambda r - 1 + \lambda r)n(x, dr) satisfies sup_{x \in D} \int (r \wedge r^2 )n(x, dr) < \infty.

December 1, 2011

On the structure of infinite dimensional Gaussian measures

Linan Che, McGill University

•  4:30-5:30pm at: McGill in Burnside Hall, room 920
Gaussian measures are often adopted in the study of integration theory in infinite dimensions due to the absence of Lebesgue measure. The general construction of infinite dimensional Gaussian measures, via “Abstract Wiener Space”(AWS), was first introduced by L. Gross and lately reformulated by D. Stroock. While AWS plays an important role in infinite dimensional integral calculus (for example, it provides a rigorous foundation for Gaussian free fields), the structure of AWS itself is of mathematical interest. Under the setting of AWS, we studied probabilistic variations of the classical Cauchy functional equation. In this process, we developed various techniques in infinite dimensional analysis which lead naturally to results about the structure of AWS. This is joint work with D. Stroock.