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

Kihoon Seong

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arXiv · 2308.01378

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

Kihoon Seong

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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.

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Kihoon Seong. 2023-08-02. Exponential ergodicity for the stochastic hyperbolic sine-Gordon equation on the circle. https://arxiv.org/abs/2308.01378

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