## 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

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

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

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.