We consider a class of 1d Lagrangian systems with random forcing in the spaceperiodic setting: $${\phi }_t+{\phi }_x^2/2=F^{\omega },x\in S^1=ℝ/\mathbb{Z}.$$ 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...