Olivier Raimond

Flows, coalescence and noise

We are interested in stationary fluid random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.

Self-interacting diffusions. III. Symmetric interactions

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to dX t =dW t (X t )-1 t( t 0 ⊇V Xs (X t )ds)dt, where {W t } is a Brownian vector field on M and V x (y) = V(x, y) a smooth function. Let μ t = 1 t t 0 δ Xs ds denote the normalized occupation measure of X t . We prove that, when V is symmetric, μ t converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μ t converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

Self-interacting diffusions. II: Convergence in law

Abstract This paper concerns convergence in law properties of self-interacting diffusions on a compact Riemannian manifold.

Flots browniens isotropes sur la sphère

Resume Nous etudions les flots browniens isotropes sur des espaces homogenes et particulierement sur la sphere Sd−1. Un flot brownien isotrope est dirige par ξ un champ de vecteurs gaussiens isotrope et est donc caracterise par une matrice de covariance. En utilisant les representations irreductibles de SO(d), nous calculons cette matrice de covariance. Connaissant cette matrice de covariance, nous calculons les exposants de Lyapounov du flot, qui decrivent son comportement asymptotique. En particulier, nous voyons que pour d ≤ 5, un flot de gradient est toujours stable.

A Class of Self-Interacting Processes with Applications to Games and Reinforced Random Walks

This paper studies a class of non-Markovian and nonhomogeneous stochastic processes on a finite state space. Relying on a recent paper by Benaim, Hofbauer, and Sorin [SIAM J. Control Optim., 44 (2005), pp. 328-348] it is shown that, under certain assumptions, the asymptotic behavior of occupation measures can be described in terms of a certain set-valued deterministic dynamical system. This provides a unified approach to simulated annealing type processes and permits the study of new models of vertex reinforced random walks and new models of learning in games such as Markovian fictitious play.

Flots de noyaux et flots coalescents

We present a part of the results of Le Jan and Raimond (math.PR/9909147). We show that starting with a compatible family of Feller semigroups, one can construct a stochastic flow of kernels. Under an additional hypotheses (on the 2-points motion), we show that it is possible to associate to a flow of kernels a coalescing flow such that the flow of kernels can be obtained by filtering the coalescing flow with respect to a sub-noise of an extension of the noise generated by the coalescing flow. To cite this article: Y. Le Jan, O. Raimond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A Bakry-Emery Criterion for Self-Interacting Diffusions

We give a Bakry-Emery type criterion for self-interacting diffusions on a compact manifold.