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Alex Cox
Email: A.M.G.Cox@bath.ac.uk

### 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 self-similar Markov processes and time-reversal properties

7/3/07: Simon Harris

Branching Brownian motion in a quadratic potential

14/3/07: Akira Sakai

Critical behavior for long-range 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 heay-tailed (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).

Parabolic Anderson model: Localisation of mass in random media

We study the parabolic Anderson problem, i.e., the heat equation on the d-dimentional integer lattice with independent identically distributed random potential and localised initial condition. Our interest is in the long-term behaviour of the random total mass of the unique non-negative 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, continuous-state branching processes and so-called generalized Flemming-Viot super-processes, 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 small-times, 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 Fleming-Viot process, such as the Wright-Fisher- or the Moran model, then the standard means to describe its genealogy is Kingman's coalescent. For this coalescent process, powerful inference methods are well-established.
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 so-called generalised $\Lambda$-Fleming-Viot processes, as well as their (dual) genealogies, given by the so-called $\Lambda$-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 sub-family of Beta $\left(2-\alpha ,\alpha \right)$-coalescents, where $\alpha$ in $\left(1,2\right]$, 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: ${M}_{n}=c\left(p\right)n-g\left(n\right)+O\left(1\right)$ with $c$ and $g$ explicitly known. But how well can we do in a limited search time? It is easy to see, by finite look-ahead rules, that the time to find a path with $\left(c\left(p\right)-x\right)n$ 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 near-critical branchingrandom walk with absorption.

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