The Dynamical Sine-Gordon Model

Abstract

We introduce the dynamical sine-Gordon equation in two space dimensions with parameter β, which is the natural dynamic associated to the usual quantum sineGordon model. It is shown that when β 2 ∈ (0, 16π 3 ) the Wick renormalised equation is well-posed. In the regime β 2 ∈ (0, 4π), the Da Prato-Debussche method [DPD02, DPD03] applies, while for3 ), the solution theory is provided via the theory of regularity structures [Hai13]. 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 [LRV13].

On the two-dimensional hyperbolic stochastic sine-Gordon equation

“…In the stationary and parabolic settings, analogous results were established by Lacoin, Rhodes, and Vargas [28, Theorem 3.1] and Hairer and Shen [24, Theorem 2.1]. 5 The main difference between Proposition 1.1 and the previous results in [28,24] is the dependence of the regularity on the time parameter T > 0. In particular, as T increases, the regularity of Θ N gets worse.…”

“…In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local wellposedness of SSG for any value of a parameter β 2 > 0 in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: 0 < β 2 < 8π. We also present a triviality result for the unrenormalized SSG.2010 Mathematics Subject Classification.…”

“…It turns out that there is an infinite number of thresholds: β 2 = j j+1 8π, j ∈ N, where one encounters new divergent stochastic objects, requiring further renormalizations. By using the theory of regularity structures [22], Hairer and Shen proved local well-posedness up to the second threshold β 2 < 16π 3 in the first work [24]. In the second work [8], together with Chandra, they pushed the local wellposedness theory to the entire subcritical regime β 2 < 8π.…”

“…setting [24,8], formally indicates existence of an infinite number of thresholds T j = T j (β), now given in terms of time,…”

Singular SPDEs in domains with boundaries

“…From the linear theory we know that u is not expected to have better (parabolic) regularity than 1/2, so its spatial derivative is a distribution, which, in general, one cannot take the square of. The theory developed in [10] provided a robust concept of solution to equations like KPZ [9], Φ 4 3 , the parabolic Anderson model in both two [10] and three [12] dimensions, the dynamical Sine-Gordon model [14] on the torus, or such equations on the whole Euclidean space [11]. As neither the torus nor the whole space has boundaries, the spatial behaviour in these examples are 'uniform', and the only blow-up of the generalised abstract Taylor expansions-also referred to as 'modelled distributions'-that describe the solutions occur at the {t = 0} hyperplane of the initial time.…”