The dynamical sine-Gordon model in the full subcritical regime

Abstract

We prove that the dynamical sine-Gordon equation on the two dimensional torus introduced in [HS16] is locally well-posed for the entire subcritical regime. At first glance this equation is far out of the scope of the local existence theory available in the framework of regularity structures [Hai14, BHZ16, CH16, BCCH17] since it involves a non-polynomial nonlinearity and the solution is expected to be a distribution (without any additional small parameter as in [FG17, HX18]). In [HS16] this was overcome by a change of variable, but the new equation that arises has a multiplicative dependence on highly non-Gaussian noises which makes stochastic estimates highly non-trivial - as a result [HS16] was only able to treat part of the subcritical regime. Moreover, the cumulants of these noises fall out of the scope of the later work [CH16]. In this work we systematically leverage "charge" cancellations specific to this model and obtain stochastic estimates that allow us to cover the entire subcritical regime.

arXiv comment64 pages

References in corpus (7)

• Renormalising SPDEs in regularity structures
• The dynamical sine-Gordon model
• Sine-Gordon Revisited
• The motion of a random string
• Critical exponents of the two dimensional Coulomb gas at the Berezinskii-Kosterlitz-Thouless transition
• Weak universality for a class of 3d stochastic reaction-diffusion models
• Large-scale limit of interface fluctuation models

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