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Keywords:
Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity
Lagrangian dynamics, Random dynamical systems, Invariant measure, Hyperbolicity
Summary:
We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: \begin{equation} \nonumber \phi_t+\phi_x^2/2=F^{\omega},\ x \in S^1=\R / \Z. \end{equation} These systems have been studied since the 1990s by Khanin, Sinai and their collaborators [7, 9, 11, 12, 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 [11]. In the multidimensional setting, a more technically involved proof has been recently given by Iturriaga, Khanin and Zhang [13].
References:
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