The dynamical sine-Gordon model

Understand

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter $\beta$, which is the natural dynamic associated to the usual quantum sine-Gordon model.

• It is shown that when $\beta^2 \in (0,\frac{16\pi}{3})$ the Wick renormalised equation is well-posed.
• In the regime $\beta^2 \in (0,4\pi)$, the Da Prato-Debussche method applies, while for $\beta^2 \in [4\pi,\frac{16\pi}{3})$, the solution theory is provided via the theory of regularity structures (Hairer 2013).
• We also show that this model arises naturally from a class of $2+1$-dimensional equilibrium interface fluctuation models with periodic nonlinearities.