Stochastic Analysis Seminar
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Mon, 11/10/2010 14:15 |
Jie Xiong (University of Tennessee) |
Stochastic Analysis Seminar  |
Eagle House |
| For a superprocess in a random environment in one dimensional space, a nonlinear stochastic partial differential equation is derived for its density by Dawson-Vaillancourt-Wang (2000). The joint continuity was left as an open problem. In this talk, we will give an affirmative answer to this problem. | |||
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Mon, 11/10/2010 15:45 |
Ben Hambly |
Stochastic Analysis Seminar |
Eagle House |
| We review the problem of determining the high frequency asymptotics of the spectrum of the Laplacian and its relationship to the geometry of a domain. We then establish these asymptotics for some continuum random trees as well as the scaling limit of the critical random graph. | |||
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Mon, 18/10/2010 14:15 |
Jeremie Unterberger |
Stochastic Analysis Seminar |
Eagle House |
| Rough path theory, invented by T. Lyons, is a successful and general method for solving ordinary or stochastic differential equations driven by irregular Hölder paths, relying on the definition of a finite number of substitutes of iterated integrals satisfying definite algebraic and regularity properties. Although these are known to exist, many questions are still open, in particular: (1) "how many" possible choices are there ? (2) how to construct one explicitly ? (3) what is the connection to "true" iterated integrals obtained by an approximation scheme ? In a series of papers, we (1) showed that "formal" rough paths (leaving aside regularity) were exactly determined by so-called "tree data"; (2) gave several explicit constructions, the most recent ones relying on quantum field renormalization methods; (3) obtained with J. Magnen (Laboratoire de Physique Theorique, Ecole Polytechnique) a Lévy area for fractional Brownian motion with Hurst index <1/4 as the limit in law of iterated integrals of a non-Gaussian interacting process, thus calling for a redefinition of the process itself. The latter construction belongs to the field of high energy physics, and as such established by using constructive field theory and renormalization; it should extend to a general rough path (work in progress). | |||
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Mon, 18/10/2010 15:45 |
Nadia Sidorova |
Stochastic Analysis Seminar |
Eagle House |
| We consider a dilute stationary system of N particles uniformly distributed in space and interacting pairwise according to a compactly supported potential, which is repellent at short distances and attractive at moderate distances. We are interested in the large-N behaviour of the system. We show that at a certain scale there are phase transitions in the temperature parameter and describe the energy and ground states explicitly in terms of a variational problem | |||
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Mon, 25/10/2010 14:15 |
Annie Millet |
Stochastic Analysis Seminar |
Eagle House |
| We consider a non linear Schrödinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak. | |||
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Mon, 25/10/2010 15:45 |
Istvan Berkes (Graz University of Technology) |
Stochastic Analysis Seminar |
Eagle House |
| The sequence {nα}, where α is an irrational number and {.} denotes fractional part, plays a fundamental role in probability theory, analysis and number theory. For suitable α, this sequence provides an example for "most uniform" infinite sequences, i.e. sequences whose discrepancy has thesmallest possible order of magnitude. Such 'low discrepancy' sequences have important applications in Monte Carlo integration and other problems of numerical mathematics. For rapidly increasing nk the behaviour of {nkα} is similar to that of independent random variables, but its asymptotic properties depend strongly also on the number theoretic properties of nk, providing a simple example for pseudorandom behaviour. Finally, for periodic f the sequence f(nα) provides a generalization of the trig-onometric system with many interesting properties. In this lecture, we give a survey of the field (going back more than 100 years) and formulate new results. | |||
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Mon, 01/11/2010 14:15 |
Martin Huesmann |
Stochastic Analysis Seminar |
Eagle House |
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Mon, 01/11/2010 15:45 |
Alison Etheridge (University of Oxford) |
Stochastic Analysis Seminar |
Eagle House |
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Mon, 08/11/2010 14:15 |
Mark Jerrum |
Stochastic Analysis Seminar |
Eagle House |
| Abstract: An instance of the Potts model is defined by a graph of interactions and a number, q, of different “spins”. A configuration in this model is an assignment of spins to vertices. Each configuration has a weight, which in the ferromagnetic case is greater when more pairs of adjacent spins are alike. The classical Ising model is the special case of q=2 spins. We consider the problem of computing an approximation to the partition function, i.e., weighted sum of configurations, of an instance of the Potts model. Through the random cluster formulation it is possible to make sense of the partition function also for non-integer q. Yet another equivalent formulation is as the Tutte polynomial in the positive quadrant. About twenty years ago, Jerrum and Sinclair gave an efficient (i.e., polynomial-time) algorithm for approximating the partition function of a ferromagnetic Ising system. Attempts to extend this result to q≠2 have been unsuccessful. At the same time, no convincing evidence has been presented to indicate that such an extension is impossible. An interesting feature of the random cluster model when q>2 is that it exhibits a first-order phase transition, while for 1≤q≤2 only a second-order phase transition is apparent. The idea I want to convey in this talk is that this first-order phase transition can be exploited in order to encode apparently hard computational problems within the model. This provides the first evidence that the partition function of the ferromagnetic Potts model may be hard to compute when q>2. This is joint work with Leslie Ann Goldberg, University of Liverpool. | |||
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Mon, 08/11/2010 15:45 |
Jan Swart |
Stochastic Analysis Seminar |
Eagle House |
| In this talk, we will look at the diffusive scaling limit of a class of one-dimensional random walks in a random space-time environment. In the scaling limit, this gives rise to a so-called stochastic flow of kernels as introduced by Le Jan and Raimond and generalized by Howitt and Warren. We will prove several new results about these stochastic flows of kernels by making use of the theory of the Brownian web and net. This is joint work with R. Sun and E. Schertzer. | |||
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Mon, 15/11/2010 14:15 |
Dimitris Cheliotis |
Stochastic Analysis Seminar |
Eagle House |
| We consider a directed random polymer interacting with an interface that carries random charges some of which attract while others repel the polymer. Such a polymer can be in a localized or delocalized phase, i.e., it stays near the interface or wanders away respectively. The phase it chooses depends on the temperature and the average bias of the disorder. At a given temperature, there is a critical bias separating the two phases. A question of particular interest, and which has been studied extensively in the Physics and Mathematics literature, is whether the quenched critical bias differs from the annealed critical bias. When it does, we say that the disorder is relevant. Using a large deviations result proved recently by Birkner, Greven, and den Hollander, we derive a variational formula for the quenchedcritical bias. This leads to a necessary and sufficient condition for disorder relevance that implies easily some known results as well as new ones. The talk is based on joint work with Frank den Hollander. | |||
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Mon, 15/11/2010 15:45 |
Hubert Lacoin |
Stochastic Analysis Seminar |
Eagle House |
| We study a simple heat-bath type dynamic for a simple model of polymer interacting with an interface. The polymer is a nearest neighbor path in Z, and the interaction is modelised by energy penalties/bonuses given when the path touches 0. This dynamic has been studied by D. Wilson for the case without interaction, then by Caputo et al. for the more general case. When the interface is repulsive, the dynamic slows down due to the appearance of a bottleneck in the state space, moreover, the systems exhibits a metastable behavior, and, after time rescaling, behaves like a two-state Markov chain. | |||
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Mon, 22/11/2010 14:15 |
Neil O’Connell |
Stochastic Analysis Seminar |
Eagle House |
| We relate the partition function associated with a certain Brownian directed polymer model to a diffusion process which is closely related to a quantum integrable system known as the quantum Toda lattice. This result is based on a `tropical' variant of a combinatorial bijection known as the Robinson-Schensted-Knuth (RSK) correspondence and is completely analogous to the relationship between the length of the longest increasing subsequence in a random permutation and the Plancherel measure on the dual of the symmetric group. | |||
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Mon, 22/11/2010 15:45 |
Xue-Mei Li |
Stochastic Analysis Seminar |
Eagle House |
| Probability measures in infinite dimensional spaces especially that induced by stochastic processes are the main objects of the talk. We discuss the role played by measures on analysis on path spaces, Sobolev inequalities, weak formulations and local versions of such inequalities related to Brownian bridge measures. | |||
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Mon, 29/11/2010 14:15 |
Beatrice De Tiliere |
Stochastic Analysis Seminar |
Eagle House |
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Mon, 29/11/2010 15:45 |
Rama Cont |
Stochastic Analysis Seminar |
Eagle House |
