Stochastic Analysis Seminar
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Mon, 23/04/2012 14:15 |
BEN LEIMKUHLER (University of Edinburgh) |
Stochastic Analysis Seminar  |
Oxford-Man Institute |
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I will discuss properties of stochastic differential equations and numerical algorithms for sampling Gibbs (i.e smooth) measures. Methods such as Langevin dynamics are reliable and well-studied performers for molecular sampling. I will show that, when the objective of simulation is sampling of the configurational distribution, it is possible to obtain a superconvergence result (an unexpected increase in order of accuracy) for the invariant distribution. I will also describe an application of thermostats to the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations |
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Mon, 23/04/2012 15:45 |
PHILIPP DOERSEK (ETH Zurich) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed. Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms. Parts of this work are joint with J. Teichmann and D. Veluscek. | |||
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Mon, 30/04/2012 14:15 |
MARTIN BARLOW (University of British Columbia) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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It is well known that electrical resistance arguments provide (usually) the best method for determining whether a graph is transient or recurrent. In this talk I will discuss a similar characterization of 'sub-diffusive behaviour' -- this occurs in spaces with many obstacles or traps. The characterization is in terms of the energy of functions in annuli. |
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Mon, 30/04/2012 15:45 |
MISHA SODIN (Tel Aviv University) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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We find the order of growth of the typical number of components of zero sets of smooth random functions of several real variables. This might be thought as a statistical version of the (first half of) 16th Hilbert problem. The primary examples are various ensembles of Gaussian real-valued polynomials (algebraic or trigonometric) of large degree, and smooth Gaussian functions on the Euclidean space with translation-invariant distribution. Joint work with Fedor Nazarov.
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Mon, 14/05/2012 14:15 |
BRUNO SCHAPIRA (University Paris-Sud) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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"The model of Vertex Reinforced Random Walk (VRRW) on Z goes back to Pemantle & Volkov, '99, who proved a result of localization on 5 sites with positive probability. They also conjectured that this was the a.s. behavior of the walk. In 2004, Tarrès managed to prove this conjecture. Then in 2006, inspired by Davis'paper '90 on the edge reinforced version of the model, Volkov studied VRRW with weight on Z. He proved that in the strongly reinforced case, i.e. when the weight sequence is reciprocally summable, the walk localizes a.s. on 2 sites, as expected. He also proved that localization is a.s. not possible for weights growing sublinearly, but like a power of n. However, the question of localization remained open for other weights, like n*log n or n/log n, for instance. In the talk I will first review these results and formulate more precisely the open questions. Then I will present some recent results giving partial answers. This is based on joint (partly still on-going) work with Anne-Laure Basdevant and Arvind Singh."
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Mon, 14/05/2012 15:45 |
JAN VAN NEERVAN (Delft University of Technology) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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Pathwise Holder convergence with optimal rates is proved for the implicit Euler scheme associated with semilinear stochastic evolution equations with multiplicative noise. The results are applied to a class of second order parabolic SPDEs driven by space-time white noise. This is joint work with Sonja Cox. |
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Mon, 21/05/2012 14:15 |
CHRISTIAN BAYER (University of Vienna) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| Based on ideas from rough path analysis and operator splitting, the Kusuoka-Lyons-Victoir scheme provides a family of higher order methods for the weak approximation of stochastic differential equations. Out of this family, the Ninomiya-Victoir method is especially simple to implement and to adjust to various different models. We give some examples of models used in financial engineering and comment on the performance of the Ninomiya-Victoir scheme and some modifications when applied to these models. | |||
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Mon, 21/05/2012 15:45 |
DEJAN VELUSCEK (ETH Zurich) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| We will give a quick overview of the semigroup perspective on splitting schemes for S(P)DEs which present a robust, "easy to implement" numerical method for calculating the expected value of a certain payoff of a stochastic process driven by a S(P)DE. Having a high numerical order of convergence enables us to replace the Monte Carlo integration technique by alternative, faster techniques. The numerical order of splitting schemes for S(P)DEs is bounded by 2. The technique of combining several splittings using linear combinations which kills some additional terms in the error expansion and thus raises the order of the numerical method is called the extrapolation. In the presentation we will focus on a special extrapolation of the Lie-Trotter splitting: the symmetrically weighted sequential splitting, and its subsequent extrapolations. Using the semigroup technique their convergence will be investigated. At the end several applications to the S(P)DEs will be given. | |||
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Mon, 28/05/2012 14:15 |
CHRISTOPHE SABOT (Universite Lyon 1) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We show that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure. The mixing measure is interpreted as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer. This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010). (Joint work with Pierre Tarrès.)
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Mon, 28/05/2012 15:45 |
HUGO DUMINIL (Unversity of Geneva) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| <> abstract:In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary. | |||
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Mon, 11/06/2012 14:15 |
KAY KIRKPATRICK (University of Illinois, Chicago) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| There are two main statistical mechanical models of ferromagnetism: the simpler and better-understood Ising model, and the more realistic and more challenging classical Heisenberg model, where the spins are in the 2-sphere instead of in {-1,+1}. In dimensions one and two, the classical Heisenberg model with nearest-neighbor interactions has no phase transition, but in three dimensions it has been intractable. To shed some light on the qualitative behavior of the 3D Heisenberg model, we use the versatile tools of mean-field theory and Stein's method in recent work with Elizabeth Meckes, studying the Heisenberg model on a complete graph with the number of vertices going to infinity. Our results include detailed descriptions of the magnetization, the empirical spin distribution, the free energy, and a second-order phase transition. | |||
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Mon, 11/06/2012 15:45 |
FREDRIK JOHANSSON VIKLUND (Colombia University) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| The Schramm-Loewner evolution (SLE(\kappa)) is a family of random fractal curves that arise in a natural way as scaling limits of interfaces in critical models in statistical physics. The SLE curves are constructed by solving the Loewner differential equation driven by a scaled Brownian motion. We will give an overview of some of the almost sure properties of SLE curves, concentrating in particular on properties that can be derived by studying the the geometry of growing curve locally at the tip. We will discuss a multifractual spectrum of harmonic measure at the tip, regularity in the capacity parameterization, and continuity of the curves as the \kappa-parameter is varied while the driving Brownian motion sample is kept fixed. This is based on joint work with Greg Lawler, and with Steffen Rohde and Carto Wong. | |||
