Jacques Franchi

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.

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.

Small Time Asymptotics for an Example of Strictly Hypoelliptic Heat Kernel

A small time asymptotics of the density is established for a simplified (non-Gaussian, strictly hypoelliptic) second chaos process tangent to the Dudley relativistic diffusion.

Brownian charges around loops

SummaryLet (Xtn) be a Poisson sequence of independent Brownian motions in ℝd,d≧3; Let ℒ be a compact oriented submanifold of ℝd, of dimensiond−2 and volume ℓ; let Φt be the sum of the windings of (Xsn, 0≦s≦t) around ℒ; then Φt/t converges in law towards a Cauchy variable of parameter ℓ/2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ℝd∖ℒ.

From Riemannian to Relativistic Diffusions

We first introduce Euclidean and Riemannian Brownian motions. Then considering Minkowski space, we present the Dudley relativistic diffusion. Finally we construct a family of covariant relativistic diffusions on a generic Lorentz manifold, the quadratic variation of which can be locally determined by the curvature (which allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time). Examples are considered, in some classical space-time models: Schwarzschild, Godel and Robertson-Walker manifolds.

Canonical Lift and Exit Law of the Fundamental Diffusion Associated with a Kleinian Group

Let \(\Gamma\) be a geometrically finite Kleinian group, relative to the hyperbolic space \(\mathbb{H} = \mathbb{H}^{d + 1}\), and let \(\delta\) denote the Hausdorff dimension of its limit set. Denote by \(\Phi\) the eigenfunction of the hyperbolic Laplacian \(\Delta\), associated with its first eigenvalue \(2{\lambda}_0 = \delta(\delta-d)\), and by \(Z^{\Phi}_t\) the associated diffussion on \(\mathbb{H}\), whose generator is \(\frac{1}{2}{\Delta}^{\Phi}: = \frac{1}{2}{\Phi}^{-1}{\Delta}\circ{\Phi}-{\lambda}_0\). We give a simple construction of \(Z^{\Phi}_{t}\) through its canonical lift to the frame bundle \({\mathcal{O}}{\mathbb{H}}\), that allows to determine directly its asymptotic behaviour.

Enroulements Browniens et Subordination dans les Groupes de Lie

Le resultat de Spitzer [22], sur l’enroulement du mouvement brownien plan autour de l’origine, a suscite de multiples travaux (cf. par exemple [15, 19, 5]). Notre propos est d’une part, par l’emploi d’une echelle de temps adequate permettant d’obtenir des resultats non asymptotiques, de faire apparâitre clairement le lien entre ce type de theoreme et l’operation de subordination, d’autre part de montrer qu’il est susceptible d’etre etendu a des groupes de Lie non abeliens. Dans les deux premieres sections, sont respectivement etudiees l’operation de subordination pour un mouvement brownien dans un groupe de Lie, et son application au processus d’enroulement. Des exemples sont presentes dans la troisieme section, notamment celui de l’enroulement dans des pointes hyperboliques complexes, qui conduit a un processus de Levy sur le groupe d’Heisenberg.