Stochastic Analysis Seminar
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Mon, 11/10/2004 14:15 |
Professor Sylvie Meleard (Universite Paris 10) |
Stochastic Analysis Seminar  |
DH 3rd floor SR |
| We are interested in a microscopic stochastic description of a population of discrete individuals characterized by one adaptive trait. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death, as influenced by each individual's trait values, and interactions between individuals. An offspring usually inherits the trait values of her progenitor, except when a mutation causes the offspring to take an instantaneous mutation step at birth to new trait values. Once this point process is in place, the quest for tractable approximations can follow different mathematical paths, which differ in the normalization they assume (taking limit on population size , rescaling time) and in the nature of the corresponding approximation models: integro or integro-differential equations, superprocesses. In particular cases, we consider the long time behaviour for the stochastic or deterministic models. | |||
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Mon, 11/10/2004 15:45 |
Professor N H Bingham (University of Sheffield) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Joe Doob, who died recently aged 94, was the last survivor of the founding fathers of probability. Doob was best known for his work on martingales, and for his classic book, Stochastic Processes (1953). The talk will combine an appreciation of Doob's work and legacy with reminiscences of Doob the man. (I was fortunate to be a colleague of Doob from 1975-6, and to get to know him well during that year.) Following Doob's passing, the mantle of greatest living probabilist descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick man. | |||
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Mon, 18/10/2004 14:15 |
Dr J Trashorras (University Paris 9) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| The Hopfield model took his name and its popularity within the theory of formal neural networks. It was introduced in 1982 to describe and implement associative memories. In fact, the mathematical model was already defined, and studied in a simple form by Pastur and Figotin in an attempt to describe spin-glasses, which are magnetic materials with singular behaviour at low temperature. This model indeed shows a very complex structure if considered in a slightly different regime than the one they studied. In the present talk we will focus on the fluctuations of the free energy in the high-temperature phase. No prior knowledge of Statistical mechanics is required to follow the talk. | |||
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Mon, 18/10/2004 15:45 |
Dr Franck Barthe (Institut de Mathematiques Laboratoire de Statistique et Probabilites, Toulouse, France) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We shall review recent progress in the understanding of isoperimetric inequalities for product probability measures (a very tight description of the concentration of measure phenomeonon). Several extensions of the classical result for the Gaussian measure were recently derived by functional analytic methods. | |||
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Mon, 25/10/2004 14:15 |
Dr J Warren (University of Warwick) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| I will consider a stochastic process ( \xi_u; u \in \Gamma_\infty ) where \Gamma_\infty is the set of vertices of an infinite binary tree which satisfies some recursion relation \xi_u= \phi(\xi_{u0},\xi_{u1}, \epsilon_u) \text { for each } u \in \Gamma_\infty. Here u0 and u1 denote the two immediate daughters of the vertex u. The random variables ( \epsilon_u; u\in \Gamma_\infty), which are to be thought of as innovations, are supposed independent and identically distributed. This type of structure is ubiquitous in models coming from applied proability. A recent paper of Aldous and Bandyopadhyay has drawn attention to the issue of endogeny: that is whether the process ( \xi_u; u \in \Gamma_\infty) is measurable with respect to the innovations process. I will explain how this question is related to the existence of certain dynamics and use this idea to develop a necessary and sufficient condition [ at least if S is finite!] for endogeny in terms of the coupling rate for a Markov chain on S^2 for which the diagonal is absorbing. | |||
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Mon, 25/10/2004 15:45 |
Professor Zhongmin Qian (Oxford) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| I will present a new formula for diffusion processes which involving Ito integral for the transition probability functions. The nature of the formula I discovered is very close to the Kac formula, but its form is similar to the Cameron-Martin formula. In some sense it is the Cameron-Martin formula for pinned diffusions. | |||
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Mon, 01/11/2004 14:15 |
Professor Y M Suhov (Cambridge) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Anderson localisation is an important phenomenon describing a transition between insulation and conductivity. The problem is to analyse the spectrum of a Schroedinger operator with a random potential in the Euclidean space or on a lattice. We say that the system exhibits (exponential) localisation if with probability one the spectrum is pure point and the corresponding eigen-functions decay exponentially fast. So far in the literature one considered a single-particle model where the potential at different sites is IID or has a controlled decay of correlations. The present talk aims at $N$-particle systems (bosons or fermions) where the potential sums over different sites, and the traditional approach needs serious modifications. The main result is that if the `randomness' is strong enough, the $N$-particle system exhibits localisation. The proof exploits the muli-scale analysis scheme going back to Froehlich, Martinelli, Scoppola and Spencer and refined by von Drefus and Klein. No preliminary knowledge of the related material will be assumed from the audience, apart from basic facts. This is a joint work with V Chulaevsky (University of Reims, France) | |||
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Mon, 01/11/2004 15:45 |
Professor Dong Zhao (Academy of Mathematics and Systems Science, Beijing) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Under the nondegenerate condition as in the diffusion case, we show that the linear stochastic jump diffusion process projected on the unite sphere has an uni que invariant probabolity measure. The Lyapunov exponentcan be represented as an integral over the sphere. These results were extended to the degenerated and Levy jump cases. | |||
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Mon, 08/11/2004 14:15 |
Dr Ken Duffy (Hamilton Institute, National University of Ireland, Maynooth) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Let {X_n} be a sequence of bounded, real-valued random variables. Assume that the partial-sums processes {S_n}, where S_n=X_1+...+X_n, satisfies the large deviation principle with a convex rate-function, I(). Given an observation of the process {X_n}, how would you estimate I()? This talk will introduce an estimator that was proposed to tackle a problem in telecommunications and discuss it's properties. In particular, recent results regarding the large deviations of estimating I() will be presented. The significance of these results for the problem which originally motivated the estimator, estimating the tails of queue-length distributions, will be demonstrated. Open problems will be mentioned and a tenuous link to Oxford's Mathematical Institute revealed. | |||
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Mon, 08/11/2004 15:45 |
Dr Jochen Voss (University of Warwick) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We present a large deviation result for the behaviour of the end-point of a diffusion under the influence of a strong drift. The rate function can be explicitely determined for both attracting and repelling drift. It transpires that this problem cannot be solved using Freidlin-Wentzel theory alone. We present the main ideas of a proof which is based on the Girsanov-Formula and Tauberian theorems of exponential type. | |||
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Mon, 15/11/2004 14:15 |
Professor Oleg Smolyanov (Moscow University) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Hamiltonian Feynman path integrals, or Feynman (path) integrals over trajectories in the phase space, are values, which some pseudomeasures, usually called Feynman (pseudo)measures (they are distributions, in the sense of the Sobolev-Schwartz theory), take on functions defined on trajectories in the phase space; so such functions are integrands in the Feynman path integrals. Hamiltonian Feynman path integrals (and also Feynman path integrals over trajectories in the configuration space) are used to get some representations of solutions for Schroedinger type equations. In the talk one plans to discuss the following problems. | |||
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Mon, 15/11/2004 14:45 |
Professor Sergei Kuksin (Heriot-Watt University, Edinburgh) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| I shall talk on recent results on behaviour of solutions of 2D Navier-Stokes Equation (and some other related equations), perturbed by a random force, proportional to the square root of the viscosity. I shall discuss some properties of the solutions, uniform in the viscosity, as well as the inviscid limit. | |||
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Mon, 22/11/2004 14:15 |
Professor Alexander Yu Veretennikov (School of Mathematics, University of Leeds) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Ergodic Markov processes possess invariant measures. In the case if transition probabilities or SDE coefficients depend on a parameter, it is important to know whether these measures depend regularly on this parameter. Results of this kind will be discussed. Another close topic is whether approximations to Markov diffusions possess ergodic properties similar to those of the limiting processes. Some partial answer to this question will be presented. | |||
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Mon, 22/11/2004 15:45 |
Dr D Crisan (Imperial College London) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| /notices/abstracts/stochastic-analysis/ht04/crisan.shtml | |||
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Mon, 29/11/2004 14:15 |
Mark Meredith (Magdalen College) |
Stochastic Analysis Seminar |
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| We propose two models of the evolution of a pair of competing populations. Both are lattice based. The first is a compromise between fully spatial models, which do not appear amenable to analytic results, and interacting particle system models, which don't, at present, incorporate all the competitive strategies that a population might adopt. The second is a simplification of the first in which competition is only supposed to act within lattice sites and the total population size within each lattice point is a constant. In a special case, this second model is dual to a branching-annihilating random walk. For each model, using a comparison with N-dependent oriented percolation, we show that for certain parameter values both populations will coexist for all time with positive probability. As a corollary we deduce survival for all time of branching annihilating random walk for sufficiently large branching rates. We also present conjectures relating to the role of space in the survival probabilities for the two populations. | |||
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Mon, 29/11/2004 15:45 |
Professor Christina Goldschmidt (University of Cambridge) |
Stochastic Analysis Seminar |
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| We describe a nice example of duality between coagulation and fragmentation associated with certain Dirichlet distributions. The fragmentation and coalescence chains we derive arise naturally in the context of the genealogy of Yule processes. | |||
