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Solving the dynamical sine-Gordon equation

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We discuss the dynamical sine-Gordon equation in two space dimension with parameter $\beta$. This is a heat equation perturbed by space-time white noise and a trigonometric nonlinearity, which is the natural dynamic associated to the sine-Gordon model in quantum field theory. We show that when $\beta2 < 16\pi /3$, the Wick renormalised equation is well-posed. In the regime $\beta2 < 4\pi$, the Da Prato-Debussche method (2003) applies, while for $\beta^2 \in [4\pi, 16\pi /3)$, the solution theory is provided via the theory of regularity structures (Hairer 2013). This is joint work with Prof. Martin Hairer.

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