Nonexplosion criteria for relativistic diffusions

Some general Lorentz covariant operators, associated to the so-called \Theta (or \Xi)-relativistic diffusions and making sense in any Lorentzian manifold, have been introduced by Franchi and Le Jan [Comm. Pure Appl. Math. 60 (2007) 187-251], Franchi and Le Jan [Curvature diffusions in general relativity (2010). Unpublished manuscript]. Only a few examples have been studied so far. We provide in this work some nonexplosion criteria for these diffusions, which can be used in generic cases.Comment: Published in at http://dx.doi.org/10.1214/11-AOP672 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

General relativistic Boltzmann equation

International audienceA new probabilistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal character of the relativistic Boltzmann model

Paracontrolled calculus and regularity structures

International audienceWe start in this work the study of the relation between the theory of regularity structures and paracontrolled calculus. We give a paracontrolled representation of the reconstruction operator and provide a natural parametrization of the space of admissible models

Diffusion in small time in incomplete sub-Riemannian manifolds

For incomplete sub-Riemannian manifolds, and for an associated second-order hy-poelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities , with optimal constant in the exponent. Under similar conditions, we obtain the small-time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric

Kinetic Brownian motion on Riemannian manifolds

International audienceWe consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T 1 M of a Riemannian manifold (M, g), collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter σ quantifying the size of the noise. Projection on M of these processes provides random C 1 paths in M. We show, both qualitively and quantitatively, that the laws of these M-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter σ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when σ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms

Propagation of chaos for mean field rough differential equations

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs

A REGULARITY STRUCTURE FOR THE QUASILINEAR GENERALIZED KPZ EQUATION (Probability Symposium)

We prove the local well-posedness of a regularity structure formulation of the quasilinear generalized KPZ equation and give an explicit form for a renormalized equation in the full subcritical regime. This is an abstract of author's work [4]

Path-dependent rough differential equations

17 pagesWe show in this work how the machinery of C^1-approximate flows introduced in our previous work "Flows driven by rough paths", provides a very efficient tool for proving well-posedness results for path-dependent rough differential equations on flows of the form d\phi = V h(dt) + F X(dt), for smooth enough path-dependent vector fields V,F = (V_1,...,V_\ell), any Holder weak geometric p-rough path X and any a-Holder path h, with a+1/p>1