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

TL;DR

This paper proves exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing, confirming part of a conjecture and extending deterministic results to stochastic PDEs.

Contribution

It establishes exponential convergence in a stochastic PDE setting for 1D Lagrangian systems with random forcing, linking to previous conjectures and deterministic analogues.

Findings

Proves exponential convergence to the stationary measure.

Confirms a part of a previously formulated conjecture.

Extends deterministic results to stochastic PDE framework.

Abstract

We prove exponential convergence to the stationary measure for a class of 1d Lagrangian systems with random forcing in the space-periodic setting: $$ \phi_t+\phi_x^2/2=F^{\omega}, x \in S^1 = \mathbb{R}/\mathbb{Z}. $$ 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.