Yves Le Jan

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.

Markov loops and renormalization

We study Poissonnian ensembles of Markov loops and the associated renormalized self-intersection local times.

Markov Paths, Loops and Fields

This is an extended version of a series of lectures given in St Flour. It includes a discussion of relations between the occupation field of Markov loops with the corresponding free field.

Permanental fields, loop soups and continuous additive functionals

A permanental field, ψ={ψ(ν),ν∈V}, is a particular stochastic process indexed by a space of measures on a set S. It is determined by a kernel u(x,y), x,y∈S, that need not be symmetric and is allowed to be infinite on the diagonal. We show that these fields exist when u(x,y) is a potential density of a transient Markov process X in S. A permanental field ψ can be realized as the limit of a renormalized sum of continuous additive functionals determined by a loop soup of X, which we carefully construct. A Dynkin-type isomorphism theorem is obtained that relates ψ to continuous additive functionals of X (continuous in t), L={Lνt,(ν,t)∈V×R+}. Sufficient conditions are obtained for the continuity of L on V×R+. The metric on V is given by a proper norm.

Curvature Diffusions in General Relativity

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.

Statistic of the winding of geodesics on a Riemann surface with finite area and constant negative curvature

In this paper we show that the windings of geodesics around the cusps of a Riemann surface of a finite area, behave asymptotically as independent Cauchy variables.

Stable windings on hyperbolic surfaces

Abstract. Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law under the Patterson-Sullivan measure on T1ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff dimension of the limit set of the subgroup Γ of Möbius isometries associated with ℳ. The normalization is t−1/(2δ−1), for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan measure mentioned above by measures that are regular along the stable leaves.

Dynkin’s isomorphism without symmetry

The purpose of this note is to extend Dynkins isomorphim involving functionals of the occupation field of a symmetric Markov processes and of the associated Gaussian field to a suitable class of non symmetric Markov processes.

Loop Measures Without Transition Probabilities

We construct Markov loop measures without assuming the existence of densities for transition probabilities.

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).