Informal Probability
Seminars: Spring 2008
Our seminars our usually held at 12.15 p.m. on
Wednesdays in room 1W 3.6 . If you
wish to find out more, please contact one of the
organisers. The speakers are mostly internal (Bath)
unless otherwise stated. Details of previous
semesters can be found
here
18/1/08: Max von Renesse(TU
Berlin)
Particle Approximation of the
Wasserstein Diffusion
The Wasserstein Diffusion is a special random process
on the probability distributions on the unit interval
which is closely related to the quadratic Wasserstein
distance and whose unique invariant measure admits a
formal Gibbs type representation with the relative
entropy as Hamiltonian.
We review the initial construction by Dirichlet form
methods first. In the second part of the talk we
present an approximation by a sequence of interacting
particle systems.
Note time and day: 11.15am,
Friday
8/2/08: Olivier Zindy(WIAS)
Aging for random walks in random
environment in the sub-ballistic regime
We consider transient one-dimensional random walks in
random environment with zero asymptotic speed. An
aging phenomenon involving the generalized Arcsine
law is proved using the localization of the walk at
the foot of `valleys' of height log t.
Note time and day: 10.15am,
Friday
13/2/08: Pierre Patie
(Bern)
Law of the exponential functional
of a family of one-sided Lévy
processes.
J. Bertoin and M. Yor determine the law of the
exponential functional of a spectrally positive
Lévy process with a negative mean through
their negative entire moments. In this talk, we start
by computing, in terms of new power series, the
Laplace transform of such a functional associated to
a spectrally negative Lévy process satisfying
Rivero's condition. Then, specifying on a new family
of one-sided Lévy processes, we provide an
expression for the density of their corresponding
exponential functionals in terms of the Wright
hypergeometric functions. This is achieved by
connecting such laws with the entrance laws of the
family of self-similar continuous state branching
processes with immigration. We end up by establishing
some interesting analytical properties enjoyed by the
Wright hypergeometric functions.
20/2/08: Peter Mörters
The local time of additive Levy
processes in supercritical dimensions
26/2/08: Markus Heydenreich
(TU/e)
Long-range percolation
I plan to introduce long-range percolation and review
selected literature. I will discuss a recent result
on mean-field behaviour of long-range percolation,
obtained jointly with Remco van der Hofstad and Akira
Sakai.
Note time and day: 11.15,
Tuesday
27/2/08: Marcel Ortgiese
Small value probabilities via the
branching tree heuristic
We present an easy and intuitive technique for
calculating small value probabilities in a wide range
of problems. Our primary example is the well-known
small value problem for the martingale limit of a
supercritical Galton-Watson process. But we will see
that the same intuition also works in the quite
different context of self-intersections of stochastic
processes. For instance, it allows us to identify the
small value probabilities for intersection local
times of several Brownian motions, as well as for the
self-intersection local times of a single Brownian
motion.
10/3/08: Nikos Zygouras (USC)
Pinning-depinning transition in
Random Polymers
Abstract
Note time and day: 11.15,
Monday
12/3/08: Janos Englander
An interacting spatial branching
model - work in progress
I will try to share a couple of ideas with the
audience about a spatial particle (branching) system
with a simple interaction. We will also study how the
center of mass behaves for certain spatial branching
systems.
28/3/08: Jean Bertoin (Paris VI)
The structure of the allelic
partition for Galton-Watson processes with neutral
mutations
Note time and day: TBA, Friday
9/4/08: Julien Berestycki (Paris VI)
Kingman coalescent and Brownian excursion : some
classical results revisited with an application to the
multifractal spectrum of Kingman's coalecsent
It is a folk theorem to say that the
genealogy of a Brownian excursion is given by
Kingman's coalescent. Two important and well known
results hint at this. Le Gall has shown that the
genealogy of the Dawson-Watanabe superprocessus is
encoded in the Brownian excursion while Perkins, in
his so-called desintegration theorem, shows that a
renormalized Dawson-Watanabe superprocessus is just a
Flemming-Viot superprocessus whose genealogy is well
known to be Kingman's coalescent. Here, we give an
explicit embedding of Kingman's coalescent in the
Brownian excursion which allows us to compute the
multifractal spectrum of the coalescent. Those results
complements an earlier joint work with N. Berestycki
and J. Schweinsberg.
16/4/08: Loïc Chaumont (Angers)
Reflection principle and Ocone
martingales
Let $M =(M_t)_{t\geq
0}$ be any continuous real-valued stochastic process. We
prove that if there exists a sequence $(x_n)$ of real
numbers which converges to 0 and such that $M$ satisfies
the reflection property at all levels $x_n$ and $2x_n$,
then $M$ is a local Ocone martingale. We state the
subsequent open question: is this result still true when
the property only holds at levels $x_n$~? Then we prove
that the later question is equivalent to the fact that
for Brownian motion, the sigma field of the invariant
events by all the reflections at levels $x_n$, $n\ge1$
is trivial.
23/4/08: Stanislav Volkov
Random walks with time and space
dependent drifts
We will consider a one-dimensional discrete-time
stochastic process with asymptotically zero drift, which
depends both on the time and the position of the
walker. We establish an interesting phase transition of
these walks, which cover a whole range of other models:
from Lamperti's problem to Friedman's urn model. For the
latter, we manage to answer an apparently still open
question.
Based on a joint work with Mikhail Menshikov (Durham)
30/4/08: Mathew Penrose
Normal approximation in stochastic
geometry via size-biased couplings
Consider n random points in a square of volume n, each
surrounded by a unit disk. We discuss the rate of normal
approximation for the number of isolated points, and for
the amount of covered area.
14/5/08: Simon Harris
Inhomogenous branching Brownian
motion : the perils of a quadratic
potential
21/5/08: Vadim Shcherbakov
On a simple growth model
A simple spread/growth model will be
defined. The asymptotic behaviour of the model has been
studied in details in a special case (typical results
will be given) and no results known (?) in a general
case, which will be briefly discussed.
16/6/08: Ben Kaehler
Pricing American Rainbow Options and
Multivariate Variance Gamma Processes
This talk is on two topics related to pricing
American rainbow options using Lévy processes. An
American rainbow option is an option on two or more
underlying assets that can be exercised at any time
before maturity. Pricing an American Rainbow option
under the usual Black-Scholes assumptions becomes
difficult in high dimensions due to the curse of
dimensionality. A method for making those calculations
tractable is discussed. Extending those results to a
world where prices follow multivariate exponential
Lévy processes requires additional assumptions
about the processes. Some alternatives for generalising
the successful Variance Gamma process into higher
dimensions are presented for that purpose.
Note time and day: 11.15, Monday
Seminars will be added to this list as they are
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