Prob-L@b
Seminars: Spring 2010
Our seminars our usually held at 12.15
p.m. on Wednesdays in room 1W 3.6
before 19/4/2010, and 4W 1.7
after this date. 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
17/2/10: Mathew Penrose
Covariance bounds and
chaos expansion over Poisson spaces
3/3/10: Antal Járai
Zero dissipation limit
in the Abelian sandpile model
10/3/10: Alexander Cox
Time-Homogeneous
Diffusions with a Given Marginal at a
Random Time
17/3/10: Gregorio
Moreno (Paris)
Asymmetric directed
polymers in random
environments
We consider the point-to-point partition
function of discrete time directed polymers
in a random environment. We show that, when
the end point is close to an axis and the
temperature is properly rescaled, the free
energy coincides with the one of a continuous
time model in a Brownian environment. This
limit is explicit in the one-dimensional case
(Moriarty-O'Connell 2007).
In a second time, we consider the
point-to-line partition function at fixed
temperature for one-dimensional directed
polymers with a drift growing with the size
of the box. Based on estimates for last
passage percolation, we compute the free
energy and give the exact order of the
fluctuations.
24/3/10: Daniel Gentner
(Karlsruher Institut für Technologie)
Inspecting partially
stationary models of Stochastic Geometry:
Theory and applications
We introduce the
notion of "generalized Palm measure" which
arises in partially stationary models where
the underlying group operation with respect
to which the model is distributionally
invariant is not transitive. Though a
minimal level of formality is inevitable
for this introduction, our focus will then
be to skip technical details and move
quickly towards applications and related
results. These include the recently quite
popular Mass-Transport Principle, extended
to non-transitive group operations, along
with a quick application demonstrating its
usefulness, a result about
isometry-stationary tesselations on compact
2-dim Riemannian manifolds and explicit
results about the distribution of "typical
cells on group invariant subsets of
$\mathds{R}^d$" in partially stationary or
isotropic Poisson Delauney
Tesselations. Here, of course, we will have
to clarify the exact meaning of such
"typical cells".
14/4/10: Wolfgang
König (Berlin)
Phase transitions for
dilute particle systems with Lennard-Jones
potential
Note venue: 3E2.1
21/4/10: Gabriel Faraud
(Paris XIII)
Random walk in random
environment, the case of
trees
The model of random walks in random
environment studied by Sinaï and
Solomon has received much interest over the
past twenty years, and is now quite well
understood. In parallel, many extensions to
this model have been introduced, and some
of them, in particular random walk in
random environment on Zd remain quite
mysterious. We will particularly focus on
the case of trees. Indeed, apart from its
theoretical interest, as a intermediate
between dimension 1 and multidimensional
models, this model presents some links with
models of branching random walk, which
allows to get quite precise results on its
behaviour. We will try to explicit what
this links are, and and give a overview of
some recent results on this topic.
Note venue: 3W3.7
28/4/10: Christian
Mönch
Dynamical Random Graphs
with Sublinear Preferential
Attachment
The
Sublinear Preferential Attachment Graph is
a dynamical network model whose evolution
in (discrete) time is governed by a
sublinear function of the degrees of its
vertices. I will give an overview of
results concerning main attributes of the
graph like degree distribution and
emergence of a giant component. If time
allows, I will also prove lower bounds on
the diameter for important special
cases. Our model is a natural
generalisation of the Barabasi-Albert model
and the results support predictions of
universality for a large class of random
networks.
4/5/10: Tom Kurtz
(Madison-Wisconsin)
Weak convergence and
large deviation theory
Puhalskii, O'Brien and Vervaat, and de
Acosta have developed an approach to large
deviation theory that is directly analogous
to the theory of weak convergence of
probability measures, including a complete
analog of Prohorov's theorem based on a
notion of exponential tightness. This
approach will be described, emphasizing the
parallels with weak convergence
theory. Necessary and sufficient conditions
for exponential tightness for a sequence of
cadlag processes in the Skorohod topology
will be given that are similar to standard
results for weak convergence.
Large deviations for
Markov processes
For a
sequence of Markov processes, convergence
of Fleming's log-exponential nonlinear
semigroups is shown to imply the large
deviation principle in a manner analogous
to the use of convergence of linear
semigroups in weak convergence. In
particular cases, this convergence can be
verified using the theory of nonlinear
contraction semigroups. The theory of
viscosity solutions of nonlinear equations
is used to generalize earlier results on
the semigroup convergence, enabling
application of the method to obtain a
variety of new and known results. Control
methods similar to methods presented in the
book by Dupuis and Ellis are used to give
representations of the rate functions.
Note time and venue: Tuesday,
10:00-12:15, 4W1.17
19/5/10: Robert Knobloch
One-sided FKPP
travelling waves for fragmentation
processes
We introduce
the FKPP equation in the setting of
fragmentation processes. Our main results
are concerned with the existence and
uniqueness of one-sided FKPP travelling
waves. In this respect we consider a
product martingale that is related to such
a travelling wave solution.
26/5/10: Jiajie Wang
Root's Barriers and
Partial Differential Equations:
Construction and Optimality
Recent work of Dupire and Carl & Lee
has emphasised the importance of
understanding the Skorokhod Embedding
originally proposed by Root for
applications in the model-free hedging of
variance options. Root's work shows that
there exists a barrier from which one may
define a stopping time which solves the
Skorokhod embedding problem. This
construction has the remarkable property,
proved by Rost, that it minimises the
variance of the stopping time among all
solutions.
In this work, we prove a characterisation
of Root's barrier in terms of the solution
to a free boundary problem, originally
stated by Dupire, and we give an
alternative proof of the optimality
property which has an important consequence
for the construction of hedging strategies
in the financial context.
2/6/10: Andreas Kyprianou
The prolific backbone
for a super-diffusion
21/6/10: Hermann
Thorisson (University of Iceland)
Mass-stationarity
through the Cox process
Consider a random measure on a
locally compact Abelian group, for instance
the d-dimensional Euclidean space. Consider
also a random element in a measurable space
on which the group acts, for instance a
random field indexed by the
group. Mass-stationarity of the random
element with respect to the measure is an
intrinsic characterization of Palm versions
with respect to stationary random
measures. It is a formalization of the
intuitive idea that the origin is a typical
location in the mass of the
measure. Mass-stationarity is an extension
to random measures of point stationarity
with respect to a simple point
process.
A Cox process represents the mass of a
random measure through a collection of
points placed independently at typical
locations in the mass. Thus if the random
measure is mass-stationary and we add an
extra point at the origin to the Cox
process then the points of that modified
Cox process are all at typical locations in
the mass of the random measure. It turns
out that mass-stationarity with respect to
the random measure reduces to
mass-stationarity with respect to the
modified Cox process. In particular, for
diffuse random measures mass-stationarity
reduces in this way to point stationarity.
Note time and day: Monday, 14:15
15/7/10: Narn-Rueih
Shieh (National Taiwan University)
The Exponential
Stationary Processes
Note time and day:
Thursday, 15.15
Seminars will be added to this list as they are
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