The dynamical sine-Gordon model (Q2634684)

In this paper, the authors aimed to solve the following periodic nonlinear SPDEs on the two-dimensional torus \(T^2\): \[ \partial_t u = \frac 1 2 \Delta u + c\sin(\beta u+\theta) + \xi_t, \] where \(c\), \(\beta\), \(\theta\) are constants and \(\xi\) is the space-time white noise. The solution is constructed by solving a regular SPDE with \(\xi\) replaced by \(\xi_\epsilon:= \xi* \rho_\epsilon\) for a smooth probability density \(\rho\) with compact support and \(\rho_\epsilon(t,x):= \epsilon^{-4}\rho(\epsilon^{-2} t, \epsilon^{-1}x)\), then letting \(\epsilon \to 0\) to approach a process which is independent of \(\rho\). More generally, consider \[ \partial_t u = \frac 1 2 \Delta u + c_\epsilon F_\beta(u_\epsilon) + \xi_\epsilon, \] where \(F_\beta\) is a trigonometric polynomial of the form \[ F_\beta(u) =\sum_{k=1}^N \zeta_k\sin(k\beta u+\theta_k) \] for \(N\in \mathbb N\) and constants \(\zeta_k, \beta, \theta_k\). As a main result, it is proved that, for \(\beta\in (0, \frac{4\pi}{3})\), \(\eta\in (-\frac{1}{3},0)\), and \(u_0\in C^\eta(T^2)\), there exists a constant \(C\), depending only on \(\beta\) and \(\rho\), such that the solution \(u_\epsilon\) to the equation for \(c_\epsilon= C\epsilon^{-\beta^2/(4\pi)}\) converges in probability to a distributional process \(u\) which is independent of the choice of \(\rho\), which is understood as a solution to the original SPDE.