Reflecting a Langevin process at an absorbing boundary

We consider a Langevin process with white noise random forcing. We suppose that the energy of the particle is instantaneously absorbed when it hits some fixed obstacle. We show nonetheless, the particle can be instantaneously reflected, and study some properties of this reflecting solution.

On ultrametricity in spin glass theory

One of the crucial issues in spin-glass theory is the supposed ultrametricity structure of the so-called "pure states". Despite of the recent progress on the Sherrington-Kirpatrick model, this is still an open question. We discuss some simple examples where ultrametricity can be proved, and which (hopefully) shed some light on the issue.
(Joint work with Nicola Kistler).

A stochastic variational derivation of the Navier-Stokes equation

A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold. Navier-Stokes equation is reinterpreted as a perturbed equation of geodesics for the L² norm.

Transition path theory

Transition Path Theory is the statistical theory of the reactive trajectories between two pre-specified sets A and B, i.e. the portions of the path of a Markov process during which the path makes a transition from A to B. We show that various probability distributions on the reactive trajectories can be expressed in terms of the equilibrium distribution of the process and the so-called committor functions which give the probability that the process reaches first B before reaching A, either backward or forward in time.

Multiplicative large deviations and turbulent transport

Many characteristics of the transport in turbulent flows important for a range of applications from chemical reactivity of admixtures to time scale for rain onset or treshold for drag reduction in polymer solutions may be related to the statistics of large deviations of the finite-time Lyapunov exponents in the flow. Examples of such characteristics include the long time behavior of the density fluctuations or multi-fractal dimensions of the attractor measures. I shall discuss how in the Kraichnan model of turbulent flow a relation to known integrable models permits to establish the existence of the multiplicative large deviation regime for the finite-time Lyapunov exponents and to calculate the corresponding rate function.

Heat kernel measures and Riemannian geometry in infinite dimensions

We will consider several different examples of infinite-dimensional manifolds: Hilbert-Schmidt groups which are natural infinite-dimensional analogues of matrix groups, loop groups and the homogeneous space Diff(S1)/S1 associated with the Virasoro algebra. We will list what is known about the Ricci curvature in each of the case, and how its boundness (or unboundenss) is reflected in the heat kernel measure behaviour.

On the derivation of Fourier law

We discuss the problem of deriving Fourier law of heat conduction for a Hamiltonian system consisting of a lattice of coupled anharmonic oscillators with noise on the boundary of the domain.

Some combinatorial aspects of Yang-Mills theory

Yang-Mills theory in two dimensions is almost a topological theory because it is invariant under area-preserving diffeomorphisms. Once it is discretized, it becomes a combinatorial theory closely related to fat graphs on surfaces. I will explain how this point of view allows one to compute rigorously and in a simple manner the partition function. I will also describe a new discrete Yang-Mills theory which, unlike the usual one, allows one to include the topology of the bundle at the discrete level. In other words, this is a discrete theory of connections on non-trivial bundles over surfaces. The continuum limit of this theory exists and provides a decomposition into mutually singular parts of the continuum limit of the usual discrete theory.

Glauder system associated with continuous gas

Brownian motion on volume preserving diffeomorphisms group and existence of global solutions of 2D stochastic Euler equation

A geometric Brownian motion performs a continuous time infinitesimal perturbation of the state of the system; this perturbation conserves the energy; exact expression for the energy transfer towards high modes is obtained ensuring existence for all time of the solution.

On the spectral theory of quantum graphs

On quantum contiguity

Lucien Le Cam introduced contiguity in Probability Theory as part of his work on the mathematical foundations of asymptotic statistical analysis. The author extended this notion to the analysis of open system dynamics (classical or quantum) in [2] and [1]. The conference is aimed at giving a survey of recent results on contiguity and its applications to Quantum Information, namely, the relation of contiguity with fidelity.

Plan

1. Introduction: Le Cam's view on contiguity.
2. Contiguity of states.
3. Møller operators and contiguity.
4. Examples from classical dynamics.
5. Examples of quantum nature.
6. Measurements and contiguity.
7.Quantum delity.

References

[1] Rolando Rebolledo. The wave map of Feller semigroups. In Stochastic analysis and mathematical physics (Santiago, 1998), Trends Math., pages 109-121. Birkähuser Boston, Boston, MA, 2000.

[2] Rolando Rebolledo. Limit problems for quantum dynamical semigroups -- inspired by scattering theory. In Quantum probability communications, Vol. XII (Grenoble, 1998), QP-PQ, XII, pages 139-172. World Sci. Publishing, River Edge, NJ, 2003.

Random walk in a Dirichlet environment

Mathematical aspects of QCD in two dimensions

Two dimensional quantum gauge theory, studied originally as a warm-up for the four dimensional theory, has proved to be rich in mathematical ideas and thereby interesting in its own right. This talk will present an overview of some of the history, central ideas, and mathematical results and directions in quantum chromodynamics in two dimensions. The quantum field theory here involves stochastic analysis in formulating the functional integrals rigorously, it involves topology and symplectics when considering the gauge field on compact surfaces, and connects up with free probability theory when the symmetry group is taken to be U(N) with large N.

On the disconnection of discrete cylinders

In this talk we will report on some recent results obtained in part with A. Dembo concerning simple random walk in a discrete cylinder with a base which is a large finite graph of bounded degree. The main focus lies in understanding the time it takes the walk to disconnect the cylinder or expressed in a more picturesque language, we consider the problem of a "termite in a wooden beam".

Some results on differential operators with random elements

The talk deals with localization of the principal value of an elliptic operator with random elements and asymptotic behavior of solution to some parabolic equations including nonlinear terms with random coefficients.