The canonical time is Tuesday afternoon, but some
talks are on other afternoons.
All interested are encouraged
to take part to the full by presenting their ideas and discussing
those of others. Graduate students especially are urged to attend.
All talks will be in Room S27 of the Laboratory, unless
otherwise announced.
Tuesday February 1
2.00pm Mathew Penrose (Durham)
Central limit theorems in spatial probability via martingales
The classical central limit theorem for martingale differences, which
dates back to Levy, has recently been shown to have fruitful applications
in showing that various functionals of a spatial white noise process
observed in a large window, are approximately normally distributed.
The method has applications to percolation, to spatial epidemics, and to
functions of spatial point patterns. We shall discuss some of these.
Tuesday February 8
2.00pm David Sheridan
Central limit theorems for graphical models
Random graphs arise naturally in many probabilistic models, including
percolation and the random cluster model, as well as the related Ising and
Potts magnets. The martingale method can be extended to prove central limit
theorems for all these models, even at points of criticality. Previous
applications of this method have not adapted to dependent measures or
any sequence of random variables whose variance grows sub-linearly. If time
permits, an alternative central limit theorem using Stein's method will be
given.
Tuesday February 15
2.00pm Geoffrey Grimmett
Exponential decay in spatial processes
When spatial processes are subcritical, they usually have correlations
which decay exponentially quickly, and this property is
key to the understanding of the subcritical phase. Exponential decay
has been proved for percolation
and certain epidemic processes. It is an open question to prove exponential
decay for the three dimensional Ising model, and for the random-cluster model
in dimensions higher than three. Exponential decay for
the existence of large entanglements is unproven, although near-exponential
decay, i.e., exponential subject to a logarithmic correction,
is known. This lecture will contain a general discussion of the available
methods and the open problems.
Tuesday February 22
2.00pm Kurt Johansson (Stockholm)
Random growth and random matrices
Tuesday February 29
2.00pm Guy Gielis
Coupled map lattices with phase transition
Robert MacKay and I have
constructed infinite systems of coupled (deterministic) maps
(CML) starting from probabilistic cellular automata (PCA) in order to
find (rigorously proven) examples of phase transition in CML.
In the talk, I will describe PCA and explain how they can be used to
build CML. The simple relation between the stochastic and the
deterministic evolution enables us to almost copy several properties of
the PCA to the CML. I will not assume any knowledge of one of the
systems.
Tuesday March 7
2.00pm Minoru Murata (Tokyo)
Uniqueness of nonnegative solutions of the Dirichlet problem
for parabolic equations in cylinders
The purpose of this talk is to give a sharp sufficient condition
for uniqueness of nonnegative solutions of the Dirichlet problem
$$
\align
\partial_tu & =Lu \qquad \text{in}\quad D \times (0,T),\\
u(x,0) & =u_0(x) \qquad \text{on}\quad D,\\
u(x,t) & =0 \qquad \text{on}\quad \partial D \times (0,T).
\endalign
$$
Here $L$ is a second order locally uniformly elliptic
partial differential operator on a
Riemannian manifold $M$,
$D$ is a domain in $M$ which is not relatively compact,
and $T$ is a positive constant.
By making use of the parabolic Harnack inequality and Carleson estimate,
we shall show that the uniqueness holds if all ends of $D$ are
(intuitively speaking) wide.
Tuesday March 14
2.00pm J. P. N. Bishwal (Calcutta)
Fractional Stochastic Calculus and Maximum Likelihood Estimation
for SDEs driven by Fractional Brownian Motion
We discuss the basic properties of the fractional Brownian motion
(fBm) and review several approaches to stochastic integration
with respect to fBm. We give conditions on existence and
uniqueness of solutions of SDEs driven by fBm and state the
fractional equivalent of Ito's formula and Girsanov's formula. Then
we study the properties of the maximum likelihood estimator
of a parameter appearing in the drift coefficient of SDEs driven
by fBm when the corresponding solution, a Dirichlet process,
is observed continuously on a finite time interval [0,T].
3.00pm Irina Kurkova (Eurandom)
Riemann surfaces and random walks in a quarter-plane
A review of the analytic theory of random walks
in a quarter plane will be given, based on the
approach by V.Malyshev.
Easter Term 2000
Tuesday May 2
2.00pm Malvina Luczak (Oxford)
Reducing network congestion and blocking probability through balanced
allocation
We compare the performance of a variant of the standard {\it Dynamic
Alternative Routing (DAR)} technique commonly used in telephone and
ATM networks to a path selection algorithm that is based on the
balanced allocations principle - the
{\it Balanced Dynamic Alternative Routing (BDAR)} algorithm.
While the standard technique checks alternative routes sequentially
until available bandwidth is found, the BDAR algorithm
compares and chooses the best among a small number of alternatives.
We show that, at the expense of a minor increase in routing overhead,
the BDAR gives a substantial improvement in network performance in
terms of both network congestion and blocking probabilities.
(Joint work with Eli Upfal)
Tuesday May 9
2.00pm Markus Kraft
Numerical study of a stochastic particle method for homogeneous
gas phase reaction
I will study a stochastic particle system that
describes homogeneous gas phase reactions
of a number of chemical species.
First I introduce the system as a Markov jump process and
discuss how relevant physical quantities are represented
in terms of appropriate random variables.
Then I show how various deterministic equations,
known from the literature, are derived from
the stochastic system in the limit when the number of
particles goes to infinity. Finally, I apply the corresponding
stochastic algorithm to a number of problems,
including practically relevant as the combustion of
methane and of heptane that are used as model-fuels for
natural gas furnaces and gas turbines, respectively.
In particular, the order of convergence with
respect to the number of simulation particles is studied, and
I illustrate the limitations of the method.
Tuesday May 23
2.00pm Ilya Goldsheid (QMW)
Tuesday May 30
2.00pm D. Khmelev (Heriot-Watt and Isaac Newton Institute)
Convergence to equilibrium in queueing
and related systems
Tuesday June 6
2.00pm Thierry Levy (Paris)
Tuesday June 13
2.00pm Wendelin Werner (Paris)
Critical exponents and conformal invariance in statistical physics
to return to Geoffrey Grimmett's home page.