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

We prove exponential convergence to the stationary measure for a class of 1d Lagrangian systems with random forcing in the space-periodic setting: $${\varphi }_t+{\varphi }_x^2\mathrm{/}2=F^{\omega },x\in S^1=\mathbb{R}\mathrm{/}\mathbb{Z}\mathrm{.}$$ This confirms a part of a conjecture formulated in [9]. Our result is a consequence (and the natural stochastic PDE counterpart) of the results obtained in [5, 7]. It is also the natural analogue of the deterministic result [11] which holds in a generic setting.