Exponential convergence to the stationary measure and hyperbolicity of the minimisers for random Lagrangian

Abstract

We consider a class of 1d Lagrangian systems with random forcing in the space-periodic setting. These systems have been studied since the 1990s by Khanin, Sinai and their collaborators  [7, 9, 11, 15].Here we give an overview of their results and then we expose our recent proof of the exponential convergence to the stationary measure [6]. This is the first such result in a classical setting, i.e. in the dual-Lipschitz metric with respect to the Lebesgue space $L_p$ for finite $p$, partially answering the conjecture formulated in \cite{GIKP05}. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin ans Zhang [13].