DH 3rd floor SR

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.

http://www.maths.ox.ac.uk/notices/events/abstracts/stochastic-analysis/…

Consider a graph with n vertices placed randomly in the unit

square, each connected by an edge to its nearest neighbour in a

south-westerly direction. For many graphs of this type, the centred

total length is asymptotically normal for n large, but in the

present case the limit distribution is not normal, being defined in

terms of fixed-point distributions of a type seen more commonly in

the analysis of algorithms. We discuss related results. This is

joint work with Andrew Wade.

We consider a system of interacting Fisher-Wright diffusions

which arise in population genetics as the diffusion limit of a spatial

particle model in which frequencies of genetic types are changing due to

migration and reproduction.

For both models the historical processes are constructed,

which record the family structure and the paths of descent through space.

For any fixed time, particle representations for the

historical process of a collection of Moran models with increasing particle

intensity and of the limiting interacting Fisher-Wright diffusions are

provided on one and the same probability space by means of Donnelly and

Kurtz's look-down construction.

It will be discussed how this can be used to obtain new

results on the long term behaviour. In particular, we give representations for

the equilibrium historical processes. Based on the latter the behaviour of

large finite systems in comparison with the infinite system is described on

the level of the historical processes.

The talk is based on joint work with Andreas Greven and Vlada

Limic.

We consider the Cahn Hilliard Equation in the line, perturbed by

the space derivative of a space--time white noise. We study the

solution of the equation when the initial condition is the

interface, in the limit as the intensity of the noise goes to zero

and the time goes to infinity conveniently, and show that in a scale

that is still infinitesimal, the solution remains close to the

interface, and the fluctuations are described by a non Markovian

self similar Gaussian process whose covariance is computed.

After a brief introduction to the basics of Rough Paths I'll

explain recent work by Peter Friz, Dan Stroock and myself proving that a

Brownian path conditioned to be uniformly close to a given smooth path

converges in distribution to that path in the Rough Path metric. The Stroock

Varadhan support theorem is an immediate consequence.

The novel part of the argument is to

obtain the estimate in a way that is independent of the particular norm used

in the Euclidean space when one defines the uniform norm on path space.