CIMI
CIMI - Trimestre EDP & Probabilités
31062 Toulouse cedex 9
Tel.: +33 (0)5.61.55.64.66
Email: PDE & Proba Trimester
Serge Cohen, Xue-Mei Li

La semaine sur la géométrie différentielle stochastique s'intéressera à différents problèmes où interviennent  des probabilités, de la géométrie et des équations aux dérivées partielles avec des dosages variés suivant les exposés. Toute personne intéressée par ces trois thèmes (ou au moins deux d'entre eux) peut nous contacter pour participer.

The Stochastic Differential Geometry week will consider various problems related to Probabilty, Geometry  and Partial Differential Equations. The balance between all these themes will be different from one talk to the other.  Everyone interested in at least two of these themes can contact us to participate.

Programme.

Xue Mei Li mini-course - Aspect of SDEs: irregular coefficients and homogeneization

Abstract: I will discuss (1) stochastic flows for SDEs with irregular coefficients and SDEs with mean fields effect. (2) Stochastic homogeneization for stochastic processes on manifolds. I will be particularly interested in two examples: the orthonormal frame bundle and the Hopf fibration.

Marc Arnaudon - Reflected Brownian flows in manifolds with boundary

Abstract: We approximate reflected Brownian flows in manifolds with boundary by families of flows of Brownian motion with large drift in a neighbourhood of the boundary. We prove convergence of damped parallel translation along Brownian motion with drift, to some damped parallel translation along reflected Brownian motion. We investigate the related properties for derivatives of semigroups.

Denis Bell - Admissible vector fields on path space, liftings and Hörmander's condition

We describe a method for constructing admissibe vector fields on the space of paths generated by a degenerate diffusion process taking values in a closed finite-dimensional manifold $M$. The construction works through lifting to the Wiener space via the Itô map. While his type of lifting played a fundamental role in Malliavin's probabilistic proof of Hörmander's theorem, Malliavin's construction does not work at the path space level. In particular, the role of Hörmander's Lie algebra condition in the path space setting is rather unclear. In this talk we attempt to address this issue.

Thierry Lévy - Brownian motion on compact matrix groups of large dimension.

Abstract: The study of the Brownian motion on compact matrix groups brings together stochastic calculus, representation theory and combinatorics in what I find to be a very enjoyable way. In this course, I will try to share this pleasure in a presentation of two asymptotic results on the distribution of the eigenvalues of the Brownian motion at a fixed time on a compact matrix group whose dimension tends to infinity. I will firstly describe the convergence, originally established by Philippe Biane, of this ditribution to a deterministic measure, which is the analogue on the unit circle of the complex plane of the Wigner semi-circle distribution on the real line. I will then discuss the fluctuations around this convergence, which were studied by Mylène Maïda and myself in the unitary case, and more recently by my former student Antoine Dahlqvist in the orthogonal and symplectic cases. We shall in particular try to understand how and why non-centered Gaussian fluctuations occur in the orthogonal and symplectic cases.

Mark-K Von Renesse - A Dirichlet Form Approach to Averaging

Abstract: We present a simple approach to the averaging phenomenon of Langevin type stochastic systems, based on Mosco convergence of non-symmetric Dirichlet forms. The limiting dynamics will be a diffusion process on connected the level sets of the Hamiltonian whose coefficients can be easily determined using the co-area formula. The method applies in arbitrary dimension and generalizes previous results bz Freidlin-Wentzell. Joint work with Florent Barret (Max Planck Instititute Leipzig)

Zongminh Qian - Some gradient estimates for the porous medium equations

Abstract: The porous medium equation has been studied as a simple example of non-linear parabolic equations which may be degenerate, and therefore the established theory can not applied directly. The famous Aronson-Bénilan gradient estimate however was established for this class of equations which motivated intensive research in the last decades. In the presence of non-zero curvature (which is the case under lower order perturbation to the porous medium equations), it remains a problem how to obtain an efficient derivative estimate. We address this problem in this talk.

Speakers

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