Communications in Mathematical Physics

, Volume 341, Issue 3, pp 933–989 | Cite as

The Dynamical Sine-Gordon Model

  • Martin Hairer
  • Hao Shen


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 [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for \({\beta^{2} \in [4\pi, \frac{16\pi}{3})}\), the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of \({2 + 1}\) -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].


Heat Kernel Admissible Graph Parabolic Scaling Piecewise Linear Path Multiplicative Gaussian Chaos 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.University of WarwickCoventryUK

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