Lundi 12 juin 2023

**14:00 Zhongkai Tao **(Univ Berkeley (Californie)) Résumé

Spectral asymptotics for kinetic Brownian motion on Riemannian manifolds

Salle B407, bâtiment B, LAGA, Institut Galilée, Université Paris 13 The kinetic Brownian motion is a stochastic process that interpolates
between the geodesic flow and Laplacian. It is also an analogue of
Bismut’s hypoelliptic Laplacian operator. I will talk about a simple
proof of the convergence of the spectrum of kinetic Brownian motion to
the spectrum of base Laplacian for all compact Riemannian manifolds
without boundary, which generalizes recent work of Kolb--Weich--Wolf on
constant curvature surfaces and is analogous to the theorem of
Bismut--Lebeau for hypoelliptic Laplacian. As an application, we prove
the optimal convergence rate of kinetic Brownian motion to the
equilibrium (given by the spectral gap of the base Laplacian)
conjectured by Baudoin--Tardif. This is based on joint work with Qiuyu
Ren.

Lundi 19 juin 2023

**14:00 **()

Séminaire Tournant IHP

IHP

Lundi 3 juillet 2023

**14:00 Angeliki Menegaki **(IHES) Résumé

Spectral gap for harmonic and weakly anharmonic chain of oscillators

Salle B407, bâtiment B, LAGA, Institut Galilée, Université Paris 13We
consider one-dimensional chains and multi-dimensional networks of
harmonic oscillators coupled to two Langevin heat reservoirs at
different temperatures. Each particle interacts with its nearest
neighbours by harmonic potentials and all individual particles are
confined by harmonic potentials, too. In previous works we investigated
the sharp N-particle dependence of the spectral gap of the associated
generator in different physical scenarios and for different spatial
dimensions. We also obtained estimates on the gap after perturbing
weakly the quadratic potentials, through a Log-Sobolev
Inequality. In this talk I will present new results on the behaviour of
the spectral gap when considering longer-range interactions in the
purely harmonic chain. In particular, depending on the strength of the
longer-range interaction, there are different regimes appearing where
the gap drastically changes behaviour but even the hypoellipticity of
the operator breaks down.

Parts of this talk are joint works with Simon Becker (ETH).