Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing

Abstract

We prove exponential convergence to the stationary measure for a class of 1d Lagrangian systems with random forcing in the space-periodic setting: $$\begin{aligned} \phi _t+\phi _x^2/2=F^{\omega }, x \in S^1=\mathbb {R}/\mathbb {Z}. \end{aligned}$$ 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. This partially answers the conjecture formulated in Gomes et al. (Moscow Math J 5:613–631, 2005). Our result is a consequence (and the natural stochastic PDE counterpart) of the results obtained in Boritchev and Khanin (Nonlinearity 26(1):65, 2013), Weinan et al. (Ann Math 151:877–960, 2000). It is also the natural analogue of the deterministic result (Iturriaga and Sanchez-Morgado in J Differ Equ 246(5):1744–1753, 2009) which holds in a generic setting.