On the two-dimensional hyperbolic stochastic sine-Gordon equation

Tadahiro Oh (1) , Tristan Robert (2) , Philippe Sosoe , Yuzhao Wang (3)

1 Edin. - University of Edinburgh
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
3 Department of Mathematics and Physics

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local well-posedness of SSG for any value of a parameter $\beta^2 > 0$ in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: $0 < \beta^2 < 8 \pi$. We also present a triviality result for the unrenormalized SSG.

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Tadahiro Oh, Tristan Robert, Philippe Sosoe, Yuzhao Wang. On the two-dimensional hyperbolic stochastic sine-Gordon equation. Stochastics and Partial Differential Equations: Analysis and Computations, 2020, ⟨10.1007/s40072-020-00165-8⟩. ⟨hal-02434797⟩