Stochastic Analysis Seminar
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Mon, 19/01/2004 14:15 |
Terry Lyons (Oxford) |
Stochastic Analysis Seminar  |
DH 3rd floor SR |
| 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. | |||
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Mon, 19/01/2004 15:45 |
Stella Brassesco (Warwick) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| 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. | |||
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Mon, 26/01/2004 14:15 |
Anita Wilson |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| 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. | |||
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Mon, 26/01/2004 15:45 |
Mathew Penrose |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| 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. | |||
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Mon, 02/02/2004 15:45 |
Enzo Orsingher (University of Rome) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| http://www.maths.ox.ac.uk/notices/events/abstracts/stochastic-analysis/h... | |||
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Mon, 09/02/2004 14:15 |
Elena Zhizhina (Moscow) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| A reveiw of results about spectral analysis of generators of some stochastic lattice models (a stochastic planar rotators model, a stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral analysis of the generator of stochastic continuous particle system. The construction of one-particle subspaces of the generators and the spectral analysis of the generator restricted on these subspaces will be the focus of the talk. | |||
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Mon, 09/02/2004 15:45 |
Martijn Pistorius (King's College, London) |
Stochastic Analysis Seminar |
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. | |||
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Mon, 16/02/2004 14:15 |
Etienne Pardoux (Universite de Provence) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| The probabilistic approach to homogenization can be adapted to fully degenerate situations, where irreducibility is insured from a Doeblin type condition. Using recent results on weak sense Poisson equations in a similar framework, obtained jointly with A. Veretennikov, together with a regularization procedure, we prove the homogenization result. A similar approach can also handle degenerate random homogenization. | |||
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Mon, 16/02/2004 15:45 |
Thomas Mountford (Ecole Polytechnique) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We discuss estimating the growth exponents for positive solutions to the random parabolic Anderson's model with small parameter k. We show that behaviour for the case where the spatial variable is continuous differs markedly from that for the discrete case. | |||
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Mon, 23/02/2004 15:45 |
Stanislav Volkov (University of Bristol) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We consider a queuing system with three queues (nodes) and one server. The arrival and service rates at each node are such that the system overall is overloaded, while no individual node is. The service discipline is the following: once the server is at node j, it stays there until it serves all customers in the queue. After this, the server moves to the "more expensive" of the two queues. We will show that a.s. there will be a periodicity in the order of services, as suggested by the behavior of the corresponding dynamical systems; we also study the cases (of measure 0) when the dynamical system is chaotic, and prove that then the stochastic one cannot be periodic either. | |||
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Mon, 01/03/2004 14:15 |
Olaf Wittich |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We consider Brownian motion on a manifold conditioned not to leave the tubular neighborhood of a closed riemannian submanifold up to some fixed finite time. For small tube radii, it behaves like the intrinsic Brownian motion on the submanifold coupled to some effective potential that depends on geometrical properties of the submanifold and of the embedding. This characterization can be applied to compute the effect of constraining the motion of a quantum particle on the ambient manifold to the submanifold. | |||
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Mon, 08/03/2004 14:15 |
Philippe Biane (Ecole Normale Superieure) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| We give a construction of Brownian motion in a Weyl chamber, by a multidimensional generalisation of Pitman's theorem relating one dimensional Brownian motion with the three dimensional Bessel process. There are connections representation theory, especially to Littelmann path model. | |||
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Mon, 08/03/2004 15:45 |
Remco van der Hofstad (Technische Universiteit Eindhoven) |
Stochastic Analysis Seminar |
DH 3rd floor SR |
| Weakly self-avoiding walk (WSAW) is obtained by giving a penalty for every self-intersection to the simple random walk path. The Edwards model (EM) is obtained by giving a penalty proportional to the square integral of the local times to the Brownian motion path. Both measures significantly reduce the amount of time the motion spends in self-intersections. The above models serve as caricature models for polymers, and we will give an introduction polymers and probabilistic polymer models. We study the WSAW and EM in dimension one. We prove that as the self-repellence penalty tends to zero, the large deviation rate function of the weakly self-avoiding walk converges to the rate function of the Edwards model. This shows that the speeds of one-dimensional weakly self-avoiding walk (if it exists) converges to the speed of the Edwards model. The results generalize results earlier proved only for nearest-neighbor simple random walks via an entirely different, and significantly more complicated, method. The proof only uses weak convergence together with properties of the Edwards model, avoiding the rather heavy functional analysis that was used previously. The method of proof is quite flexible, and also applies to various related settings, such as the strictly self-avoiding case with diverging variance. This result proves a conjecture by Aldous from 1986. This is joint work with Frank den Hollander and Wolfgang Koenig. | |||
