Stochastic Analysis Seminar
|
Mon, 17/01/2005 14:15 |
Dr James Norris (University of Cambridge) |
Stochastic Analysis Seminar  |
DH 3rd floor SR |
| According to the Stokes-Einstein law, microscopic particles subject to intense bombardment by (much smaller) gas molecules perform Brownian motion with a diffusivity inversely proportion to their radius. Smoluchowski, shortly after Einstein's account of Brownian motion, used this model to explain the behaviour of a cloud of such particles when, in addition their diffusive motion, they coagulate on collision. He wrote down a system of evolution equations for the densities of particles of each size, in particular identifying the collision rate as a function of particle size. We give a rigorous derivation of (a spatially inhomogeneous generalization of) Smoluchowski's equations, as the limit of a sequence of Brownian particle systems with coagulation on collision. The equations are shown to have a unique, mass-preserving solution. A detailed limiting picture emerges describing the ancestral spatial tree of particles making up each particle in the current population. The limit is established at the level of these trees. | |||
|
Mon, 17/01/2005 15:45 |
Professor Enrique Andjel (Universite de Provence) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Given a countable set of sites S an a transition matrix p(x,y) on that set, we consider a process of particles evolving on S according to the following rule: each particle waits an exponential time and then jumps following a Markov chain governed by p(x,y); the particle keeps jumping until it reaches an empty site where it remains for another exponential time. Unlike most interacting particle systems, this process fails to have the Feller property. This causes several technical difficulties to study it. We present a method to prove that certain measures are invariant for the process and exploit the Kolmogorov zero or one law to study some of its unusual path properties. | |||
|
Mon, 24/01/2005 14:15 |
Dr Benedict Haas (Department of Statistics, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Fragmentation processes model the evolution of a particle that split as time goes on. When small particles split fast enough, the fragmentation is intensive and the initial mass is reduced to dust in finite time. We encode such fragmentation into a continuum random tree (CRT) in the sense of Aldous. When the splitting times are dense near 0, the fragmentation CRT is in turn encoded into a continuous (height) function. Under some mild hypotheses, we calculate the Hausdorff dimension of the CRT, as well as the maximal H | |||
|
Mon, 24/01/2005 15:45 |
Professor Stanislov Smirnov (Royal Institute of Technology, Stockholm) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| It became apparent during the last decade that in several questions in classical complex analysis extremal configurations are fractal, making them very difficult to attack: it is not even clear how to construct or describe extremal objects. We will argue that the most promising approach is to consider conformally self-similar random configurations, which should be extremal almost surely. | |||
|
Mon, 31/01/2005 14:15 |
Dr Moez Draief (Department of Statistics, Cambridge) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| /notices/events/abstracts/stochastic-analysis/ht05/draief.shtml | |||
|
Mon, 31/01/2005 15:45 |
Dr Matthias Winkel (Department of Statistics, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| It is well-known that the only space-time scaling limits of Galton-Watson processes are continuous-state branching processes. Their genealogical structure is most explicitly expressed by discrete trees and R-trees, respectively. Weak limit theorems have been recently established for some of these random trees. We study here a Markovian forest growth procedure that allows to construct the genealogical forest of any continuous-state branching process with immigration as an a.s. limit of Galton-Watson forests with edge lengths. Furthermore, we are naturally led to continuous forests with edge lengths. Another strength of our method is that it yields results in the general supercritical case that was excluded in most of the previous literature. | |||
|
Mon, 07/02/2005 14:15 |
Professor Kevin Burrage (Oxford University Computing Laboratory) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| /notices/events/abstracts/abstracts/stochastic-analysis/ht05/burrage.shtml | |||
|
Mon, 07/02/2005 15:45 |
Dr Tom Schmitz (Department of Mathematics, Switzerland) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We introduce conditions in the spirit of $(T)$ and $(T')$ of the discrete setting, that imply, when $d \geq 2$, a law of large numbers with non-vanishing limiting velocity (which we refer to as 'ballistic behavior') and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we consider the class of diffusions where the diffusion matrix is the identity, and give a concrete criterion on the drift term under which the diffusion in random environment exhibits ballistic behavior. | |||
|
Mon, 14/02/2005 14:15 |
Dr Ben Hambly (The Mathematical Institute, Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
|
Mon, 14/02/2005 15:45 |
Professor Boguslaw Zegarlinski (Imperial College, London) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
|
Mon, 21/02/2005 14:15 |
Dr Nadia Sidorova (Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| It is known that a continuous path of bounded variation can be reconstructed from a sequence of its iterated integrals (called the signature) in a similar way to a function on the circle being reconstructed from its Fourier coefficients. We study the radius of convergence of the corresponding logarithmic signature for paths in an arbitrary Banach space. This convergence has important consequences for control theory (in particular, it can be used for computing the logarithm of a flow)and the efficiency of numerical approximations to solutions of SDEs. We also discuss the nonlinear structure of the space of logarithmic signatures and the problem of reconstructing a path by its signature. | |||
|
Mon, 21/02/2005 15:45 |
Professor Michael Schroeder (University of mannheim) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| This talk attempts to survey key aspects of the mathematics that has been developed in recent years towards an explicit understanding of the structure of exponential functionals of Brownian motion, starting with work of Yor's in the 1990s | |||
|
Mon, 07/03/2005 14:15 |
Dr Gordon Blower (University of Lancaster) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
|
Mon, 07/03/2005 15:45 |
Professor Ilya Goldshield (Queen Mary University, London) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
