Abstract
We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary Gaussian multiplicative chaos, we prove local well-posedness of SSG for any value of a parameter beta ^2 > 0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (Commun Math Phys 341(3):933–989, 2016) and Chandra et al. (The dynamical sine-Gordon model in the full subcritical regime, arXiv:1808.02594 [math.PR], 2018), where the parameter is restricted to the subcritical range: 0< beta ^2 < 8 pi . We also present a triviality result for the unrenormalized SSG.
Highlights
For the spatial dimension d ≥ 2, the stochastic convolution is not a classical function but is merely a Schwartz distribution
This causes an issue in making sense of powers k and a fortiori of the full nonlinearity uk in (1.2), necessitating a renormalization of the nonlinearity
Our main goal in this paper is to extend the analysis to SNLW with a non-polynomial nonlinearity, of which trigonometric functions are the simplest
Summary
Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany. As in the case of a polynomial nonlinearity, a proper renormalization needs to be introduced to our problem. This can be seen from the regularity of the stochastic convolution as above. The main new difficulty comes from the non-polynomial nature of the nonlinearity, which makes the analysis of the relevant stochastic object non-trivial (as compared to the previous work [19]). After introducing a time-dependent renormalization, we show that the regularity of the main stochastic terms depends on both the parameter β ∈ R\{0} and the time t > 0. 16π β2 corresponds to the critical value, after which we do not expect to be able to extend the dynamics. It is quite intriguing that the singular nature of the problem (1.1) depends sensitively on time and gets worse over time, contrary to the parabolic setting
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