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Invariant Gibbs dynamics for the dynamical sine-Gordon model. (English) Zbl 1473.35363

Summary: In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter \(\beta^2 > 0\), and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range \(0 < \beta^2 < 4 \pi\) via the variational approach due to N. Barashkov and M. Gubinelli [Duke Math. J. 169, No. 17, 3339–3415 (2020; Zbl 07292332)]. We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range \(0 < \beta^2 < 2 \pi\). Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range \(0 < \beta^2 < 4 \pi\).

MSC:

35L71 Second-order semilinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 07292332
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