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Forthcoming events in this series
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15:45
Thoughts about the transition function of jump-type Markov processes
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Branching diffusion on Lobachevsky space with variable fission: the Hausdorff dimension of the limiting set
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A new look at limits theorms for sequential Monte-Carlo Methods
Abstract
/notices/events/abstracts/stochastic-analysis/ht06/Moulines.shtml
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Limit theorems for subsequences of random variables
Abstract
/notices/events/abstracts/stochastic-analysis/ht06/bobkov.shtml
15:45
A Feynman-Kac representation formula for fully nonlinear PDE's
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New estimates for the bottom of the negative spectrum of Schrodinger operators
15:45
Dyadic Parametrization of non-rectifiable curves
Abstract
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).
14:15
Multifractal aspects of Beta coalescence and stable random trees.
Abstract
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.
15:45
Numerical integration of stochastic differential equations with nonglobally Lipschitz coefficients
Abstract
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.
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Global stochastic flows without global Lipschitz conditions
Abstract
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.
15:45
New Extensions of Ito's Formula and Two-parameter p,q-variation Integrations of Local Times
Abstract
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Structure of Pareto sets in multiple objective Markov Decision Processes
Abstract
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.
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Markov operators and spectral measures of orthogonal polynomials ensembles
Abstract
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.
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Laplace method for heat processes on loop spaces via rough paths
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Invariant Measure of Numerical Solutions of SDE with Markovian Switching
Abstract
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.
15:45
Fractal Properties of Levy Trees
Abstract
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).
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Heat kernels of Schr
Abstract
I will present two-sided estimates for the heat kernel of the elliptic
Schr
15:45
Lattice gases and the Lov
Abstract
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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The Universality Classes in the Parabolic Anderson Model
Abstract
/notices/events/abstracts/stochastic-analysis/mt05/m
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Self-interacting Random Walks
Abstract
A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.
The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.
Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.
The purpose of our talk is to present our recent results on the subject [1,3,4,6].
Bibliography
[1] M. Bena
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A Markov History of Partial Observations
Abstract
Numerous physical systems are justifiably modelled as Markov processes. However,
in practical applications the (usually implicit) assumptions concerning accurate
measurement of the system are often a fair departure from what is possible in
reality. In general, this lack of exact information is liable to render the
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On some first passage problems for 1/2 semi-stable Markov processes enjoying the time-inversion property
Abstract
We review the analytic transformations allowing to construct standard bridges from a semistable Markov process, with indec 1/2, enjoying the time inversion property. These are generalized and some of there properties are studied. The new family maps the space of continuous real-valued functions into a family which is the topic of our focus. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these by the considered semi-stable process
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Queues, Directed Percolation and Random Matrices
Abstract
When two single server queues have the same arrivals process, this is said to be a `fork-join queue'. In the case where the arrivals and service processes are Brownian motions, the queue lengths process is a reflecting Brownian motion in the nonnegative orthant. Tan and Knessl [1996] have given a simple explicit formula for the stationary distribution for this queueing system in a symmetric case, which they obtain as a heavy traffic limit of the classical discrete model. With this as a starting point, we analyse the Brownian model directly in further detail, and consider some related exit problems.