Past Stochastic Analysis Seminar

The talk will describe recent collaborative work between the speaker and Professor Sanjoy Mitter of MIT on connections between continuous-time nonlinear filtering theory, and nonequilibrium statistical mechanics. The study of nonlinear filters from a (Shannon) information- theoretic viewpoint reveals two flows of information, dubbed 'supply' and 'dissipation'. These characterise, in a dynamic way, the dependencies between the past, present and future of the signal and observation processes. In addition, signal and nonlinear filter processes exhibit a number of symmetries, (in particular they are jointly and marginally Markov), and these allow the construction of dual filtering problems by time reversal. The information supply and dissipation processes of a dual problem have rates equal to those of the original, but with supply and dissipation exchanging roles. The joint (signal-filter) process of a nonlinear filtering problem is unusual among Markov processes in that it exhibits one-way flows of information between components. The concept of entropy flow in the stationary distribution of a Markov process is at the heart of a modern theory of nonequilibrium statistical mechanics, based on stochastic dynamics. In this, a rate of entropy flow is defined by means of time averages of stationary ergodic processes. Such a definition is inadequate in the dynamic theory of nonlinear filtering. Instead a rate of entropy production can be defined, which is based on only the (current) local characteristics of the Markov process. This can be thought of as an 'entropic derivative'. The rate of entropy production of the joint process of a nonlinear filtering problem contains an 'interactive' component equal to the sum of the information supply and dissipation rates. These connections between nonlinear filtering and statistical mechanics allow a certain degree of cross- fertilisation between the fields. For example, the nonlinear filter, viewed as a statistical mechanical system, is a type of perpetual motion machine, and provides a precise quantitative example of Landauer's Principle. On the other hand, the theory of dissipative statistical mechanical systems can be brought to bear on the study of sub-optimal filters. On a more philosophical level, we might ask what a nonlinear filter can tell us about the direction of thermodynamic time.    

 

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http://sag.maths.ox.ac.uk/samath/semars/abstract-plovdiv.pdf

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TBA

http://sag.maths.ox.ac.uk/samath/seminars/Sturm.pdf

TBA

/samath/seminars/njacob_abstract.pdf

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We will be discussing limit behaviour of sums along subsequences of a given sequence of non-correlated random variables. Some results are applied to the classical trigonometric system in the Berkes model. /notices/events/abstracts/stochastic-analysis/ht06/bobkov.shtml    

Using the dyadic parametrization of curves, and elementary theorems and

probability theory, examples are constructed of domains having bad properties on

boundary sets of large Hausdorff dimension (joint work with F.D. Lesley).

Lambda-coalescents were introduced by Pitman in (1999) and Sagitov (1999). These processes describe the evolution of particles that

undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Lambda has the Beta$(2-\alpha,\alpha)$ they are also known to describe the genealogies of large populations where a single individual can produce a large number of offsprings. Here we use a recent result of Birkner et al. (2005) to prove that Beta-coalescents can be embedded in continuous stable random trees, for which much is known due to recent progress of Duquesne and Le Gall. This produces a number of results concerning the small-time behaviour of Beta-coalescents. Most notably, we recover an almost sure limit theorem for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the infinite site frequency spectrum associated with mutations in the context of population genetics.

Stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients

possessing unique solutions make up a very important class in applications. For

instance, Langevin-type equations and gradient systems with noise belong to this

class. At the same time, most numerical methods for SDEs are derived under the

global Lipschitz condition. If this condition is violated, the behaviour of many

standard numerical methods in the whole space can lead to incorrect conclusions.

This situation is very alarming since we are forced to refuse many effective

methods and/or to resort to some comparatively complicated numerical procedures.

We propose a new concept which allows us to apply any numerical method of weak

approximation to a very broad class of SDEs with nonglobally Lipschitz

coefficients. Following this concept, we discard the approximate trajectories

which leave a sufficiently large sphere. We prove that accuracy of any method of

weak order p is estimated by $\varepsilon+O(h^{p})$, where $\varepsilon$ can be

made arbitrarily small with increasing the radius of the sphere. The results

obtained are supported by numerical experiments. The concept of rejecting

exploding trajectories is applied to computing averages with respect to the

invariant law for Langevin-type equations. This approach to computing ergodic

limits does not require from numerical methods to be ergodic and even convergent

in the nonglobal Lipschitz case. The talk is based on joint papers with G.N.

Milstein.

A classical result due to Kunita says that if the coefficients are global

Lipschitzian, then the s.d.e defines a global flow of homeomorphisms. In this

talk, we shall prove that under suitable growth on Lipschitz constants, the sde

define still a global flow.

In this talk I will present recent development of the generalized Ito's formulae of continuous semimartingales for less smooth function

First of all, I intend to remind us of several properties of

polyhedral cones and cone-generated orders which will be used for constructing Pareto sets in multiple objective optimisation problems.

Afterwards, I will consider multiple objective discounted Markov Decision Process. Methods of Convex Analysis and the Dynamic Programming Approach allow one to construct the Pareto sets and study their properties. For instance, I will show that in the unichain case, Pareto sets for different initial distributions are topologically equivalent. Finally, I will present an example on the optimal management of a deteriorating system.

We examine the classical orthogonal polynomial ensembles using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. Equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Applications to sharp deviation inequalities on largest eigenvalues are discussed.

Stochastic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Although the numerical methods for stochastic differential equations (SDEs) have been well studied, there are few results on the numerical solutions for SDEwMSs. The main aim of this talk is to investigate the invariant measure of numerical solutions of SDEwMSs and discuss their convergence.

Levy trees are random continuous trees that are obtained as

scaling limits of Galton-Watson trees. Continuous tree means here real tree, that is a certain class of path-connected metric spaces without cycles. This class of random trees contains in particular the continuum random tree of Aldous that is the limit of the uniform random tree with N vertices and egde length one over the square root of N when N goes to infinity. In this talk I give a precise definition of the Levy trees and I explain some interesting fractal properties of these trees. This talk is based on joint works with J-F Le Gall and M. Winkel available on arxiv : math.PR/0501079 (published in

PTRF) math.PR/0509518 (preprint)

math.PR/0509690 (preprint).

I will present two-sided estimates for the heat kernel of the elliptic

Schr

Given a family of independent events in a probability space, the probability

that none of the events occurs is of course the product of the probabilities

that the individual events do not occur. If there is some dependence between the

events, however, then bounding the probability that none occurs is a much less

trivial matter. The Lov

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