Exponential ergodicity for the stochastic hyperbolic sine-Gordon equation on the circle

TL;DR

This paper proves exponential convergence to a unique invariant measure for the stochastic hyperbolic sine-Gordon equation on the circle, despite challenges posed by its hyperbolic dynamics and lack of strong Feller property.

Contribution

It establishes exponential ergodicity and uniqueness of the invariant measure for the stochastic hyperbolic sine-Gordon equation, introducing methods to handle non-strong Feller dynamics.

Findings

Unique invariant Gibbs measure identified

Exponential convergence in Wasserstein distance proven

Overcomes challenges of non-strong Feller hyperbolic dynamics

Abstract

In this paper, we show that the Gibbs measure of the stochastic hyperbolic sine-Gordon equation on the circle is the unique invariant measure for the Markov process. Moreover, the Markov transition probabilities converge exponentially fast to the unique invariant measure in a type of 1-Wasserstein distance. The main difficulty comes from the fact that the hyperbolic dynamics does not satisfy the strong Feller property even if sufficiently many directions in a phase space are forced by the space-time white noise forcing. We instead establish that solutions give rise to a Markov process whose transition semigroup satisfies the asymptotic strong Feller property and convergence to equilibrium in a type of Wasserstein distance.