Prob-L@b
Seminars: Spring 2011
Our seminars our usually held at 12.15
p.m. on Wednesdays in room 4W
1.7. 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
9/2/11: Ben Hambly
(Oxford)
Spectral properties and
random walk on the critical random graph
16/2/11: Markus Riedle
(Manchester)
Cylindrical Lévy
processes in Banach spaces
The objective of this
talk is the introduction of cylindrical
Lévy processes and their stochastic
integral in Banach spaces.
The
degree of freedom of models in infinite
dimensions is often reflected by the
request that each mode along a dimension is
independently perturbed by the noise. In
the Gaussian setting, this leads to the
cylindrical Wiener process including from a
model point of view the very important
possibility to model a Gaussian noise in
both time and space in a great flexibility
(space-time white noise). Up to very
recently, there has been no analogue for
Lévy processes.
Based on the
theory of cylindrical processes and
cylindrical measures we introduce
cylindrical Lévy processes as a
natural generalisation of cylindrical
Wiener processes. We continue to
characterise the distribution of
cylindrical Lévy processes by a
cylindrical version of the
Lévy-Khintchine formula.
The
cylindrical approach enables the definition
of a stochastic integral with respect to a
cylindrical Lévy process without any
geometric constraints on the underlying
Banach space. We use this integral to
develop a theory of cylindrical stochastic
Cauchy problems and demonstrate its
practicalness by presenting some basic
facts on the cylindrical Ornstein-Uhlenbeck
process driven by a cylindrical Lévy
process.
(part of this talk is based on
joint work with D. Applebaum)
23/2/11: Peter
Mörters
Distances in ultrasmall
random networks
Scale free networks
are ubiquitous in our lives, appearing in
the form of social or communication
networks, collaboration networks, or the
world-wide web. One of the central
observations in the theory of scale-free
random networks is that in the case of
power-law exponents τ ∈
(2,3) networks are ultrasmall, which
means that the distance of two randomly
chosen nodes in the giant component of a
graph with n vertices is of
asymptotic order log log n. We
refine this observation and argue that
these networks can be further divided into
two universality classes corresponding to
quite different internal architecture. The
talk is based on joint work with Steffen
Dereich (Marburg) and Christian Moench
(Bath).
1/3/11: Ron Doney
(Manchester)
First passage times for
random walks and Lévy processes
The behaviour of the
tail of the distribution of the first
passage time over a fixed level has been
known for many years, but until recently
little was known about the behaviour of the
probability mass function or density
function. In this talk we describe recent
results of Vatutin and Wachtel, Doney, and
Doney and Rivero which give such
information whenever the random walk or
Lévy process is asymptotically stable.
Note day and venue: Tuesday,
3W 3.7
9/3/11: Daisuke
Shiraishi (Kyoto)
Random walks on
the random walk trace
Let S be a simple
random walk starting at the origin in
Z4. We consider G = S[0,∞)
to be a random subgraph of the integer
lattice and assume that a resistance of
unit 1 is put on each edge of the graph
G. Let Rn be the effective
resistance between the origin and
Sn. We derive the exact value of
the resistance exponent; more precisely, we
prove that n-1 E(Rn)
(log n)-1/2. Furthermore, we
derive the precise exponent for the heat
kernel of a random walk on G at the
quenched level. These results give the
answer to the problem raised by Burdzy and
Lawler (1990) in four dimensions.
23/3/11: Martin Kolb
(Oxford)
Spectral Analysis of
Diffusions with Jump Boundary
In the talk we
describe recent results concerning spectral
properties of so called diffusions with
jump boundary. Though these processes and
their generators already have been
constructed by Feller, the investigation of
their large time behaviour is much more
recent. In the talk results of
Pinsky/Ben-Ari, Leung/Li/Rakesh and
Kolb/Wuebker will be presented
29/3/11: Pierre Patie
(Université Libre de
Bruxelles)
Exponential Functional
of Lévy Processes
The law of the
exponential functional of Lévy
processes plays a prominent role from both
theoretical and applied perspectives. We
start this talk by describing some reasons
motivating its study and we review all
known results concerning the distribution
of this random variable. We proceed by
describing a new factorization identity for
the law of the exponential functional under
very mild conditions on the underlying
Lévy process. As by-product, we
provide some interesting distributional
properties enjoyed by this random variable
as well as some new analytical expressions
for its distribution (Joint work with
J.C. Pardo (CIMAT, Mexico) and M. Savov
(The University of Oxford, UK)).
Note time, day and
venue: Tuesday, 14.15, 6E 2.2
30/3/11: Curdin Ott
Russian Options with a
Cap
We are interested in a
modification of the optimal stopping
problem associated with the pricing of
Russian options in financial markets whose
underlying process is an exponential
spectrally negative Lévy process
(Ann. App. Prob., 2004, Vol. 14, No.1,
pp. 215-238). It turns out that the optimal
stopping boundary can be characterised by a
simple ordinary differential equation
involving the scale functions associated
with the given Lévy process. In
particular, it is non-flat and varies
according to the path variation of the
Lévy process. As an application, we
determine (in terms of scale functions) the
fair price of Russian options with a cap,
that is, Russian options whose payoff is
bounded from above by some constant.
6/4/11: Yogeshwaran
Dhandapani (École Normale
Supérieure)
Percolation and
directionally convex ordering of point
processes.
In this talk, we
explain the relation between directionally
convex ordering of point processes and
percolation. Directionally convex ordering
has been used to compare point processes
with same mean intensities. We will start
with a primer on directionally convex
ordering of point processes and examples of
point processes that are directionally
convex ordered. We link directionally
convex ordering to percolation as well as
clustering by showing that they impact
negatively the capacity functionals of
their corresponding Boolean models. This is
used to show ordering of some new critical
radii which act as upper and lower bounds
to the usual critical radius for
percolation of a point process. The upper
bound increases with dcx order while the
lower bound decreases.
In the second part, we exploit the fact
that many probabilities of additive
shot-noise fields of point processes can be
bounded by their Laplace transforms. This
for sparse point processes (i.e, lesser
than Poisson point process in dcx order)
can be bounded by the corresponding Laplace
transform of Poisson-driven shot-noise
fields. For a nice class of functionals,
one can compute the latter explicitly to
ascertain non-triviality of phase
transition in various percolation
models. We carry out such a program for
providing uniform upper and lower bounds
(uniform over all sparse point processes)
for the critical radius for k-percolation
and percolation in SINR
(Signal-to-Interference-Noise-Ratio)
graphs.
Note venue: 1W 2.5
4/5/11: Albert
Ferreiro-Castilla (Barcelona)
Inversion of analytic
characteristic functions and infinite
convolutions of exponentials
11/5/11: Xiong Jin(St Andrews)
Dimension result for
two-dimensional multiplicative cascade
processes
I will present a
Hausdorff dimension result for the image of
two-dimensional multiplicative cascade
processes, and obtain from this result a
KPZ-type formula which normally has one
point of phase transition.
Note venue: 3W 4.7
12/5/11: Günter
Last (Karlsruhe)
Minicourse I: Point
processes and random measures
Minicourse: Topics in Stochastic Geometry
Stochastic geometry aims to develop and to
analyze mathematical models for random
spatial patterns. This mini-course
provides an introduction into some topics
of stochastic geometry. The lectures will
not only cover some classical results for
stationary tessellations and the (Poisson)
Boolean model but will also present some
new developments on invariant transports of
random measures and continuum percolation.
Note time and venue:
16.15, 3W 3.7
13/5/11: Günter
Last (Karlsruhe)
Minicourse II: Random Tesselations
Note time and venue:
16.15, 3W 3.7
18/5/11: Günter
Last (Karlsruhe)
Minicourse III: Random
partitions and balanced invariant
transports
Minicourse IV: The
Boolean model
Note time:
10.15-12.15
18/5/11: Thomas Dusquene
(Paris VI)
General Growth
processes of trees
We introduce the
notion of a hereditary property for
discrete trees and study associated forest
growth processes hence providing a unified
approach to various reduction and growth
procedures of Galton-Watson trees that have
been studied previously. We shall prove
that the only possible limits are
Lévy trees. This combinatorial
approach provide an easy characterisation
of Lévy trees.
This is a
joint work with Matthias Winkel.
19/5/11: Günter
Last (Karlsruhe)
Minicourse V:
Percolation on planar tessellations
Note time:
16.15
25/5/11: Neil O'Connell (Warwick)
Generalizations of
Pitman's 2M-X theorem and their
applications
Pitman's 2M-X theorem
states that if Xt is a standard
one-dimensional Brownian motion and
Mt = maxs ≤ t
Xs then 2M-X is a
three-dimensional Bessel process. This
theorem has vast generalizations. I will
describe some of these including a
particular generalization involving
exponential functionals of Brownian motion
which was discovered by Matsumoto and Yor,
and a multi-dimensional version of this
which is related to the quantum Toda
lattice (this will be explained) and has
applications to a Brownian directed polymer
model.
8/6/11: Sergey Bocharov
Branching Brownian
Motion with branching at the origin
We shall discuss a
Branching Brownian Motion model, where
branching takes place at the origin. In
this model particles split at rate β
on the local time scale. We shall present
results about the almost sure asymptotic
population growth as well as the spatial
spread of the system.
14/6/11: Elie Aidekon
(TU Eindhoven)
The extremal process of
the branching Brownian motion
We look at the
branching Brownian motion on the real
line. Particles move independently
according to a Brownian motion and split
into two at exponential times. At time t,
we are interested in the 'landscape' seen
from the particle located at the minimum of
this process. What does the point process
consisting of the particles around the
minimum look like? We prove a convergence
in law of this process, and give a
description in terms of a decorated Poisson
point process. Joint work with
J. Berestycki, E. Brunet and Z. Shi.
Note day:
Tuesday
15/6/11: Samuel Cohen (Oxford)
Nonlinear expectations
and BSDEs in general probability spaces
Much work has been done on time-consistent
risk-averse decision making. One approach
is to axiomatically define a filtration
consistent nonlinear expectation, which is
a family of operators satisfying many of
the properties of the conditional
expectation, but can be nonlinear. This
axiomatic approach raises the question of
how to construct these expectations -- a
key approach is to define them using
solutions to Backward Stochastic
Differential Equations (BSDEs), but which
nonlinear expectations can be defined in
this way?
We consider nonlinear expectations in
probability spaces satisfying only the
usual conditions and separability. We give
a form of BSDE in these spaces, and show
existence and uniqueness of solutions, and
also a comparison theorem. We demonstrate
that all nonlinear expectations satisfying
a domination assumption can be expressed as
the solutions to BSDEs with Lipschitz
continuous drivers where the comparison
theorem holds, extending the results of
Coquet, Hu, Memin and Peng (2002) to
general probability spaces.
6/7/11: Piotr Milos (Warsaw)
CLT for an
Ornstein-Uhlenbeck branching system
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