Past Stochastic Analysis Seminar

26 January 2004
15:45
Mathew Penrose
Abstract
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.
  • Stochastic Analysis Seminar
26 January 2004
14:15
Anita Wilson
Abstract
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.
  • Stochastic Analysis Seminar
19 January 2004
15:45
Stella Brassesco
Abstract
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.
  • Stochastic Analysis Seminar
19 January 2004
14:15
Terry Lyons
Abstract
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.
  • Stochastic Analysis Seminar
1 December 2003
14:15
Abstract
In this talk, we consider a class of non-linear stochastic partial differential equations. We represent its solutions as the weighted empirical measures of interacting particle systems. As a consequence, a simulation scheme for this class of SPDEs is proposed. There are two sources of error in the scheme, one due to finite sampling of the infinite collection of particles and the other due to the Euler scheme used in the simulation of the individual particle motions. The error bound, taking into account both sources of error, is derived. A functional limit theorem is also derived. The results are applied to nonlinear filtering problems. This talk is based on joint research with Kurtz.
  • Stochastic Analysis Seminar
17 November 2003
15:45
Nadia Sidorova
Abstract
We construct and study different surface measures on the space of paths in a compact Riemannian manifold embedded into the Euclidean space. The idea of the constructions is to force a Brownian particle in the ambient space to stay in a small neighbourhood of the manifold and then to pass to the limit. Finally, we compare these surface measures with the Wiener measure on the space of paths in the manifold.
  • Stochastic Analysis Seminar

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