Informal Probability
Seminars: Spring 2007
Our seminars our usually held at 11.15 a.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
10/1/07: Erik Baurdoux
The McKean stochastic game driven
by a spectrally negative Lévy
process.
17/1/07: Loic Chaumont (Paris
6)
On the genealogy of conditioned
stable Lévy forests
31/1/07: Alex Cox
Pathwise inequalities of the local
time: applications to Skorokhod embeddings and
optimal stopping
7/2/07: Peter Mörters
Upper tails for intersection local
times of random walks in supercritical
dimensions
21/2/07: Juan Carlos Pardo
Weak convergence of positive
selfsimilar Markov processes and timereversal
properties
7/3/07: Simon Harris
Branching Brownian motion in a
quadratic potential
14/3/07: Akira Sakai
Critical behavior for longrange
oriented percolation
21/3/07: Søren Asmussen
(Aarhus)
RESTART: Tail Probabilities for a
Computer Reliability Problem
A job like the execution of a computer program or the
transfer of a file has an execution time with
distribution F. The job may fail at a time
distributed according to G, and is then restarted. If
H is the distribution of the total execution time
allowing for multiple restarts, what can be said
about the tail of H?
The answer is that H is always heaytailed (except if
F has a finite support). The precise form of the tail
comes out from a delicate balance between the tails
of F and G, and is presented in the logarithmic
asymptotics form familiar from large deviations
theory.
Joint work with Lester Lipsky, Robert Sheahan
(Storrs, Connecticut) and Pierre Fiorini (Portland,
Maine).
2/5/07: Nadia Sidorova
Parabolic Anderson model:
Localisation of mass in random media
We study the parabolic Anderson problem, i.e., the
heat equation on the ddimentional integer lattice
with independent identically distributed random
potential and localised initial condition. Our
interest is in the longterm behaviour of the random
total mass of the unique nonnegative solution, and
we prove the complete localisation of mass for
potentials with polynomial tails.
9/5/07: Marcel Ortgiese
Directed polymers in random
environment
** NOTE VENUE ** 3W 3.7
16/5/07: ** Rough Paths Double Bill **
Rough Paths Double Bill
11.15: Nadia Sidorova: Introduction to Rough Paths
13.15: Steffen Dereich: A support theorem and a LDP for flows generated by Kunita SDEs via rough path analysis
23/5/07: Julien Berestycki (Provence)
The speed of coalescence
Bertoin and Le Gall have recently remarked that there is a simple
bijective transformation that maps the measure
$\Lambda $ of a $\Lambda $coalescent onto the branching mechanism
$\Psi $ of a continuum branching process such that Schweinsberg's criterion on
$\Lambda $ for coming down from infinity in the coalescence
corresponds exactly to Grey's criterion for extinction in finite time for the
branching process. However no probabilistic explanation of this fact was known.
By shedding more light on the relations (which have been the focus of much attention
lately) between $\Lambda $coalescent, continuousstate branching processes and socalled generalized
FlemmingViot superprocesses, we are able to give such an explanation, and along
the way we obtain an almost sure asymptotic formula for the number of blocks at
smalltimes, which until now was known only in very specific cases.
29/6/07: Jochen Blath (Berlin)
$\Lambda $coalescents in population genetics: Are they really out there?
One of the central problems in mathematical genetics is the inference
of evolutionary parameters of a population (such as the mutation rate)
based on the observed genetic types in a finite DNA sample. If the
population model under consideration is in the domain of attraction of
the classical FlemingViot process, such as the WrightFisher or the
Moran model, then the standard means to describe its genealogy is
Kingman's coalescent. For this coalescent process, powerful inference
methods are wellestablished.
An important feature of the above class of models is, roughly
speaking, that the number of offspring of each individual is small
when compared to the total population size, and hence all ancestral
collisions are binary only. Recently, more general population models
have been studied, in particular in the domain of attraction of
socalled generalised $\Lambda $FlemingViot processes, as well
as their (dual) genealogies, given by the socalled
$\Lambda $coalescents, which allow multiple collisions. Moreover,
Eldon and Wakeley (2006) provide evidence that such more general
coalescents might actually be more adequate to describe real
populations with extreme reproductive behaviour, in particular many
marine species. In this talk, we extend methods of Ethier and
Griffiths (1987) and Griffiths and Tavaré (1994, 1995) to obtain a
likelihood based inference method for general
$\Lambda $coalescents.
In particular, we obtain a method to compute (approximate) likelihood
surfaces for the observed type probabilities of a given sample. We
argue that within the (vast) family of $\Lambda $coalescents, the parametrisable subfamily of Beta
$(2\alpha ,\alpha )$coalescents,
where $\alpha $ in
$(1,2]$, are of particular relevance.
We illustrate our method using simulated and real datasets.
This is joint work with Matthias Birkner (WIAS Berlin)
13/7/07: Robin Pemantle (Pennsylvania)
The best path in a tree is hard to find
Label the vertices of a binary tree with IID Bernoulli
$\left(p\right)$
random variables for some $p\le 1/2$. The maximum number of ones,
${M}_{n}$,
on any path of length n from the root is well understood:
${M}_{n}=c\left(p\right)ng\left(n\right)+O\left(1\right)$
with $c$ and
$g$
explicitly known. But how well can we do in a limited search time? It is easy to see, by
finite lookahead rules, that the time to find a path with
$\left(c\right(p)x)n$
ones is of order $ng\left(x\right)$ for some function $g$. I will discuss how badly $g$ blows up at zero. When $p$
is strictly less than $1/2$,
the answer is "pretty badly". More precise information is related to survival probabilities
for nearcritical branchingrandom walk with absorption.
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