On the two-dimensional hyperbolic stochastic sine-Gordon equation

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.

1 Introduction

1.1 Stochastic sine-Gordon equation

We consider the following hyperbolic stochastic sine-Gordon equation on $\mathbb {T}^2 = (\mathbb {R}/2\pi \mathbb {Z})^2$ with an additive space-time white noise forcing:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u + (1- \Delta ) u + \gamma \sin (\beta u) = \xi \\ (u, \partial _tu) |_{t = 0} = (u_0, u_1), \end{array}\right. } \quad (t, x) \in \mathbb {R}_+\times \mathbb {T}^2, \end{aligned}$$

(1.1)

where $\gamma$ and $\beta$ are non-zero real numbers and $\xi (t, x)$ denotes a (Gaussian) space-time white noise on $\mathbb {R}_+\times \mathbb {T}^2$. In this paper, we are interested in studying the model (1.1), where the linear dynamics is given by the wave equation.¹

The stochastic nonlinear wave equations (SNLW) have been studied extensively in various settings; see [11, Chapter 13] for the references therein. In recent years, there has been a rapid progress on the theoretical understanding of SNLW with singular stochastic forcing. In [19], Gubinelli, Koch, and the first author studied the following stochastic nonlinear wave equations with an additive space-time white noise on $\mathbb {T}^2$:

$$\begin{aligned} \partial _t^2 u + (1- \Delta ) u + u^k = \xi , \end{aligned}$$

(1.2)

where $k\ge 2$ is an integer. The main difficulty of this problem comes from the roughness of the space-time white noise $\xi$, which can already be seen at the linear level. Let $\Psi$ denote the stochastic convolution, solving the linear stochastic wave equation:

$$\begin{aligned} \partial _t^2 \Psi + (1- \Delta ) \Psi = \xi . \end{aligned}$$

(1.3)

For the spatial dimension $d \ge 2$, the stochastic convolution $\Psi$ is not a classical function but is merely a Schwartz distribution. See Sect. 1.2 and Lemma 2.6 below. This causes an issue in making sense of powers $\Psi ^k$ and a fortiori of the full nonlinearity $u^k$ in (1.2), necessitating a renormalization of the nonlinearity. In [19], by introducing an appropriate time-dependent renormalization, the authors proved local well-posedness of (a renormalized version of) (1.2) on $\mathbb {T}^2$. See [20, 21, 33, 35, 37, 39] for related works on nonlinear wave equations with singular stochastic forcing and/or rough random initial data. We point out that these works handle polynomial nonlinearities.

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. 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 $\Psi$ as above. More informally, we can see the necessity of a renormalization from the fast oscillation of $\sin (\beta u)$ due to the (expected) roughness of u, which makes the nonlinearity $\sin (\beta u)$ tend to 0 in some limiting sense. See Proposition 1.4. In order to counterbalance such decay and have a non-trivial solution, we will take $\gamma \rightarrow \infty$ in (1.1). See Sect. 1.2.

The main new difficulty comes from the non-polynomial nature of the nonlinearity, which makes the analysis of the relevant stochastic object particularly non-trivial (as compared to the previous work [19]). In particular, after introducing a time-dependent renormalization, we show that the regularity of the main stochastic terms depends on both the parameter $\beta \in \mathbb {R}{\setminus } \{0\}$ and the time $t>0$. See Proposition 1.1. This is a striking difference from the polynomial case.

The sine nonlinearity in (1.1) is closely related to models arising from both relativistic and quantum field theories [4, 13, 40] and has attracted a lot of attention over the past years. Moreover, the following (deterministic) one-dimensional sine-Gordon equation:

$$\begin{aligned} \partial _t^2 u - \partial _x^2 u + \sin u = 0 \end{aligned}$$

(1.4)

is known to be completely integrable, to which a vast literature was devoted. Here, we simply mention the work by McKean [31, 32]. See the references therein. In particular, in [32], McKean constructed an invariant Gibbs measure for (1.4).

In the one-dimensional case, the stochastic convolution $\Psi$ in (1.3) has (spatial) regularity $\frac{1}{2} - \varepsilon$, $\varepsilon > 0$, and thus a straightforward computation yields local well-posedness of (1.1). The situation becomes much more delicate in the two-dimensional setting. In the parabolic setting, Hairer and Shen [24] and Chandra et al. [8] studied the following parabolic sine-Gordon model on $\mathbb {T}^2$:

$$\begin{aligned} \partial _tu - \tfrac{1}{2} \Delta u + \gamma \sin (\beta u) = \xi . \end{aligned}$$

(1.5)

In particular, they observed that the difficulty of the problem depends sensitively on the value of $\beta ^2 > 0$. At a heuristic level, this model is comparable to various models appearing in constructive quantum field theory, where the dimension d can be expressed in terms of the parameter $\beta$; for example, the $\Phi ^3_d$-model (and the $\Phi ^4_d$-model, respectively) formally corresponds to (1.5) with $d = 2+ \frac{\beta ^2}{2\pi }$ (and $d = 2+ \frac{\beta ^2}{4\pi }$, respectively). In terms of the actual well-posedness theory, the Da Prato–Debussche trick [10] along with a standard Wick renormalization yields local well-posedness of (1.5) for $0< \beta ^2 <4\pi$. It turns out that there is an infinite number of thresholds: $\beta ^2 = \frac{j}{j+1}8\pi$, $j\in \mathbb {N}$, where one encounters new divergent stochastic objects, requiring further renormalizations. By using the theory of regularity structures [22], Hairer and Shen [24] proved local well-posedness up to the second threshold $\beta ^2<\frac{16\pi }{3}$ in the first work. In the second work [8], together with Chandra, they pushed the local well-posedness theory to the entire subcritical regime $\beta ^2<8\pi$. When $\beta ^2 = 8 \pi$, the Eq. (1.5) is critical and falls outside the scope of the current theory. In fact, it is expected that, for $\beta ^2 \ge 8 \pi$, any reasonable approach would yield a trivial solution, solving the linear stochastic heat equation. The Eq. (1.5) possesses a formally invariant Gibbs measure, and thus it is expected that Bourgain’s invariant measure argument [6, 7] would allow one to extend the local-in-time dynamics constructed in [8, 24] globally in time. We mention a recent work [29] on the construction of a Gibbs measure for the sine-Gordon model with a log-correlated Gaussian process (but their result is restricted to $d = 1$). We also mention recent papers [14, 43] on dynamical problems with an exponential nonlinearity in the two-dimensional setting; in [14], Garban studied the dynamical problem with an exponential nonlinearity in the parabolic setting, while in [43], Sun and Tzvetkov considered a dispersion generalized nonlinear wave equation with an exponential nonlinearity in the context of random data well-posedness theory.²

Our model (1.1) in the hyperbolic setting is sensitive to the value of $\beta ^2 >0$ as in the parabolic case. Furthermore, due to the non-conservative nature of the problem,³ the renormalization we employ is time-dependent as in [19] and the difficulty also depends on the time parameter $t>0$. See Proposition 1.1. On the one hand, by taking $t>0$ small, we can make sure that the relevant stochastic object is not too rough, allowing us to establish local well-posedness of (1.1) for any value of $\beta ^2 > 0$ (Theorem 1.2). On the other hand, even if we start the dynamics with small $\beta ^2 > 0$, our analysis, when compared to the parabolic setting [8, 24], formally indicates existence of an infinite number of thresholds $T_j = T_j(\beta )$, now given in terms of time,

$$\begin{aligned} T_j = \frac{16j\pi }{(j+1)\beta ^2}, \quad j \in \mathbb {N}, \end{aligned}$$

where we encounter new divergent stochastic objects, requiring further renormalizations. As in the parabolic case, the time $T_\infty = \frac{16 \pi }{ \beta ^2}$ corresponds to the critical value, after which we do not expect to be able to extend the dynamics.⁴ 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.

1.2 Renormalization of the nonlinearity

In order to explain the renormalization procedure, we first consider the following regularized equation for (1.1):

$$\begin{aligned} \partial _t^2 u_N + (1- \Delta ) u_N + \gamma {\hbox {Im}} \big (e^{i\beta u_N}\big ) = \mathbf {P}_N \xi , \end{aligned}$$

(1.6)

where $\mathbf {P}_N$ is a smooth frequency projector onto the (spatial) frequencies $\{n\in \mathbb {Z}^2:|n|\le N\}$, associated with a Fourier multiplier

$$\begin{aligned} \chi _N(n) = \chi \big (N^{-1}n\big ) \end{aligned}$$

(1.7)

for some fixed non-negative function $\chi \in C^\infty _c(\mathbb {R}^2)$ with ${{\,\mathrm{supp}\,}}\chi \subset \{\xi \in \mathbb {R}^2:|\xi |\le 1\}$ and $\chi \equiv 1$ on $\{\xi \in \mathbb {R}^2:|\xi |\le \tfrac{1}{2}\}$.

We first define the truncated stochastic convolution $\Psi _N = \mathbf {P}_N \Psi$, solving the truncated linear stochastic wave equation:

$$\begin{aligned} \partial _t^2 \Psi _N + (1-\Delta )\Psi _N = \mathbf {P}_N \xi \end{aligned}$$

(1.8)

with the zero initial data. With $\langle \,\cdot \, \rangle = (1+|\cdot |^2)^\frac{1}{2}$, let S(t) denote the linear wave propagator

$$\begin{aligned} S(t) = \frac{\sin (t\langle \nabla \rangle )}{\langle \nabla \rangle }, \end{aligned}$$

(1.9)

defined as a Fourier multiplier operator. Namely, we set

$$\begin{aligned} S(t) f = \sum _{n \in \mathbb {Z}^2 } \frac{\sin (t\langle n \rangle )}{\langle n \rangle } \widehat{f} (n) e_n, \end{aligned}$$

where $\widehat{f}(n)$ is the Fourier coefficient of f and $e_n(x)= (2\pi )^{-1}e^{in\cdot x}$ as in (1.26). Then, the truncated stochastic convolution $\Psi _N$, solving (1.8), is given by

$$\begin{aligned} \Psi _N (t) = \int _0^tS(t - t')\mathbf {P}_NdW(t'), \end{aligned}$$

(1.10)

where W denotes a cylindrical Wiener process on $L^2(\mathbb {T}^2)$:

$$\begin{aligned} W(t){\mathop {=}\limits ^{\text {def}}} \sum _{n \in \mathbb {Z}^2 } B_n (t) e_n \end{aligned}$$

(1.11)

and $\{ B_n \}_{n \in \mathbb {Z}^2}$ is defined by $B_n(0) = 0$ and $B_n(t) = \langle \xi , \mathbf {1}_{[0, t]} \cdot e_n \rangle _{ t, x}$. Here, $\langle \cdot , \cdot \rangle _{t, x}$ denotes the duality pairing on $\mathbb {R}\times \mathbb {T}^2$ and thus we formally have

$$\begin{aligned} B_n(t) = \langle \xi , \mathbf {1}_{[0, t]} e_n \rangle _{t, x} = \text {``}\int _0^t \int _{\mathbb {T}^2} \overline{e_n(x)} \xi ( dx dt')\text {''}. \end{aligned}$$

As a result, we see that $\{ B_n \}_{n \in \mathbb {Z}^2}$ is a family of mutually independent complex-valued⁵ Brownian motions conditioned so that $B_{-n} = \overline{B_n}$, $n \in \mathbb {Z}^2$. By convention, we normalized $B_n$ such that $\text {Var}(B_n(t)) = t$. Then, for each fixed $x \in \mathbb {T}^2$ and $t \ge 0$, we see that $\Psi _N(t,x)$ is a mean-zero real-valued Gaussian random variable with variance

$$\begin{aligned} \begin{aligned} \sigma _N(t)&{\mathop {=}\limits ^{\text {def}}} \mathbb {E}\big [\Psi _N(t,x)^2\big ] = \frac{1}{4\pi ^2}\sum _{n \in \mathbb {Z}^2} \chi _N^2(n) \int _0^t \bigg [\frac{\sin ((t - t')\langle n \rangle )}{\langle n \rangle } \bigg ]^2 dt'\\&= \frac{1}{4\pi ^2}\sum _{n \in \mathbb {Z}^2} \chi _N^2(n) \bigg \{ \frac{t}{2 \langle n \rangle ^2} - \frac{\sin (2 t \langle n \rangle )}{4\langle n \rangle ^3 }\bigg \}\sim t \log N \end{aligned} \end{aligned}$$

(1.12)

for all $t\in [0,1]$ and $N \gg 1$. We emphasize that the variance $\sigma _N(t)$ is time-dependent. For any $t > 0$, we see that $\sigma _N(t) \rightarrow \infty$ as $N \rightarrow \infty$, showing that $\{\Psi _N(t)\}_{N \in \mathbb {N}}$ is almost surely unbounded in $W^{0, p}(\mathbb {T}^2)$ for any $1 \le p \le \infty$. See also Lemma 2.6 below.

If we were to proceed with a Picard iteration to study (1.6), the next term we need to study is

$$\begin{aligned} e^{i\beta \Psi _N} = \sum _{k= 0}^\infty \frac{(i\beta )^k}{k!}\Psi _N^k. \end{aligned}$$

(1.13)

As pointed above, the power $\Psi _N^k$, $k \ge 2$, does not have any nice limiting behavior as $N \rightarrow \infty$. As in [19], we now introduce the Wick renormalization:

$$\begin{aligned} :\!\Psi _N^k(t,x)\!:\, {\mathop {=}\limits ^{\text {def}}} H_k\big (\Psi _N(t,x);\sigma _N(t)\big ) \end{aligned}$$

(1.14)

to each power $\Psi _N^k$ appearing in (1.13). Here, $H_k$ denotes the $k\hbox {th}$ Hermite polynomial, defined through the generating function:

$$\begin{aligned} e^{t x-\frac{\sigma }{2} t^2} = \sum _{k= 0}^\infty \frac{t^k}{k!}H_k(x;\sigma ). \end{aligned}$$

(1.15)

From (1.14) and (1.15), the renormalized complex exponential is then given by

$$\begin{aligned} \begin{aligned} \Theta _N(t,x)&= \,:\!e^{i\beta \Psi _N(t,x)}\!:\,\, {\mathop {=}\limits ^{\text {def}}}\sum _{k= 0}^\infty \frac{(i\beta )^k}{k!}:\!\Psi _N^k(t, x)\!:\\&= e^{\frac{\beta ^2}{2} \sigma _N(t)}e^{i\beta \Psi _N(t,x)}. \end{aligned} \end{aligned}$$

(1.16)

Following [8], we refer to $\Theta _N$ as the imaginary Gaussian multiplicative chaos. The following proposition establishes the regularity and convergence property of the imaginary Gaussian multiplicative chaos $\Theta _N$.

Proposition 1.1

Let $\beta \in \mathbb {R}{\setminus } \{0\}$ and $T>0$ such that $\beta ^2 T < 8 \pi$. Then, given any finite $p, q\ge 1$ and any $\alpha > 0$ satisfying $\alpha > \frac{\beta ^2T}{8\pi }$, the sequence of random variables $\Theta _N$ is a Cauchy sequence in $L^p(\Omega ;L^q([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2)))$ and hence converges to some limit $\Theta$ in $L^p(\Omega ;L^q([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2)))$.

In view of the convergence of the truncated stochastic convolution $\Psi _N$ to $\Psi$ (see Lemma 2.6), it is natural to write the limit $\Theta$ as

$$\begin{aligned} \Theta = \, :\!e^{i\beta \Psi }\!:. \end{aligned}$$

See Remark 1.3 below on the uniqueness of the limiting process $\Theta$.

In the stationary and parabolic settings, analogous results were established by Lacoin et al. [28, Theorem 3.1] and Hairer and Shen [24, Theorem 2.1].⁶ The main difference between Proposition 1.1 and the previous results in [24, 28] is the dependence of the regularity on the time parameter $T>0$. In particular, as T increases, the regularity of $\Theta _N$ gets worse. On the other hand, for fixed $\beta \in \mathbb {R}{\setminus } \{0\}$, by taking $T>0$ small, we can take $\{\Theta _N(t) \}_{N\in \mathbb {N}}$ almost bounded in $L^2(\mathbb {T}^2)$, $0 < t \le T$.

The proof of Proposition 1.1 is in the spirit of [24, 28]. In the case of a polynomial nonlinearity [19, 20], it is enough to estimate the second moment and then invoke the Wiener chaos estimate (see, for example, Lemma 2.5 in [20]) to obtain the $p\hbox {th}$ moment bound, since the stochastic objects in [19, 20] all belong to Wiener chaoses of finite order. The imaginary Gaussian multiplicative chaos $\Theta _N$ in (1.16), however, does not belong to any Wiener chaos of finite order. This forces us to estimate all its higher moments by hand. In Sect. 2, we present a proof of Proposition 1.1. While we closely follow the argument in [24], our argument is based on an elementary calculation. See Lemma 2.5.

1.3 Main results

In view of the previous discussion, we are thus led to study the following renormalized stochastic sine-Gordon equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u_N +(1-\Delta ) u_N + \gamma _N\sin (\beta u_N) = \mathbf {P}_N \xi \\ (u_N, \partial _tu_N) |_{t = 0} = (u_0, u_1), \end{array}\right. } \end{aligned}$$

(1.17)

where $\gamma _N$ is defined by

$$\begin{aligned} \gamma _N(t,\beta ) = e^{\frac{\beta ^2}{2}\sigma _N(t)} \longrightarrow \infty , \end{aligned}$$

(1.18)

as $N \rightarrow \infty$. We now state the local well-posedness result.

Theorem 1.2

Let $\beta \in \mathbb {R}{\setminus }\{0\}$ and $s > 0$. Given any $(u_0, u_1) \in \mathcal {H}^s(\mathbb {T}^2) = H^{s}(\mathbb {T}^2) \times H^{s-1}(\mathbb {T}^2)$, the Cauchy problem (1.17) is uniformly locally well-posed in the following sense; there exists $T_0 = T_0\big (\Vert (u_0,u_1)\Vert _{\mathcal {H}^s}, \beta \big )>0$ such that given any $0 < T\le T_0$ and $N \in \mathbb {N}$, there exists a set $\Omega _N(T)\subset \Omega$ such that

• (i) for any $\omega \in \Omega _N(T)$, there exists a unique solution $u_N$ to (1.17) in the class

  $$\begin{aligned} \Psi _N + X^\sigma (T) \subset C([0, T]; H^{-\varepsilon }(\mathbb {T}^2)), \end{aligned}$$

  for any small $\varepsilon > 0$, where $\Psi _N$ is as in (1.10), $X^\sigma (T)$ is the Strichartz space defined in (3.2), and $\sigma = \min (s, 1-\varepsilon )$,

• (ii) there exists a uniform estimate on the probability of the complement of $\Omega _N(T)$:

  $$\begin{aligned} P(\Omega _N(T)^c) \longrightarrow 0, \end{aligned}$$

  uniformly in $N \in \mathbb {N}$, as $T \rightarrow 0$,

Furthermore, there exist an almost surely positive stopping time $\tau = \tau \big (\Vert (u_0,u_1)\Vert _{\mathcal {H}^s}, \beta \big )$ and a stochastic process u in the class

$$\begin{aligned} \Psi + X^\sigma (T) \subset C([0, T]; H^{-\varepsilon }(\mathbb {T}^2)) \end{aligned}$$

for any $\varepsilon > 0$ such that, given any small $T >0$, on the event $\{ \tau \ge T\}$, the solution $u_N$ to (1.17) converges in probability to u in $C([0, T]; H^{-\varepsilon }(\mathbb {T}^2))$.

Note that, in Theorem 1.2 above, we can take the parameter $\beta ^2 > 0$ arbitrarily large. This exhibits sharp contrast with the parabolic case studied in [8, 24], where the parameter was restricted to the subcritical range: $0< \beta ^2 < 8 \pi$.

The main point is that Proposition 1.1 shows that the imaginary Gaussian multiplicative chaos $\Theta _N$ is almost a function (namely, $\alpha > 0$ small in Proposition 1.1) by taking $T = T(\beta )>0$ sufficiently small. In particular, we can apply the analysis from [19] on the polynomial nonlinearity via the Da Prato–Debussche trick to study (1.17); namely write a solution $u_N$ to (1.17) as

$$\begin{aligned} u_N = \Psi _N + v_N. \end{aligned}$$

In view of (1.16) and (1.18), the residual term $v_N$ then satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2v_N + (1-\Delta )v_N + {\hbox {Im}}\big (\Theta _N e^{i\beta v_N}\big )=0\\ (v_N,\partial _tv_N)\big |_{t=0}=(u_0,u_1). \end{array}\right. } \end{aligned}$$

(1.19)

Then, by taking $T = T(\beta ) >0$ sufficiently small, thus guaranteeing the regularity of $\Theta _N$ via Proposition 1.1, the standard Strichartz analysis as in [19] along with the fractional chain rule (Lemma 3.2) suffices to conclude Theorem 1.2. As in [19], our argument shows that u constructed in Theorem 1.2 has a particular structure $u = \Psi + v$, where $v \in X^\sigma (T)$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2v + (1-\Delta )v + {\hbox {Im}}\big (\Theta e^{i\beta v}\big )=0\\ (v,\partial _tv)\big |_{t=0}=(u_0,u_1), \end{array}\right. } \end{aligned}$$

(1.20)

where $\Theta$ is the limit of $\Theta _N$ constructed in Proposition 1.1.

Next, we consider the stochastic sine-Gordon equation (1.1) without renormalization by studying its frequency-truncated version⁷:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u_N +(1-\Delta ) u_N + \sin (\beta u_N) = \mathbf {P}_N \xi \\ (u_N, \partial _tu_N) |_{t = 0} = (u_0, u_1). \end{array}\right. } \end{aligned}$$

(1.21)

In studying the limit as $N \rightarrow \infty$, we expect the solution $u_N$ to become singular. As a result, in view of faster and faster “oscillation”, we expect $\sin (\beta u_N)$ to tend to 0 as a space-time distribution. This is the reason why we needed to insert a diverging multiplicative constant $\gamma _N$ in the renormalized model (1.17).

Remark 1.3

In Proposition 1.1 and Theorem 1.2 above, we used a smooth frequency projector $\mathbf {P}_N$ with the multiplier $\chi _N$ in (1.7). As in the parabolic case, it is possible to show that the limiting process $\Theta$ of $\Theta _N$ in Proposition 1.1 and the limit u of $u_N$ in Theorem 1.2 are independent of the choice of the smooth cutoff function $\chi$. See [34] for such an argument in the wave case (with a polynomial nonlinearity). Moreover, we may also proceed by smoothing via a mollification and obtain analogous results. In this case, the limiting processes agree with those constructed in Proposition 1.1 and Theorem 1.2.

In the following, we study the limiting behavior of $u_N$, solving the unrenormalized model (1.21) with regularized noises, and establish a triviality result. The heuristics above indicates that the nonlinearity $\sin (\beta u_N)$ tends to 0 in some suitable sense, indicating that $u_N$ converges to a solution to the linear stochastic wave equation. The next proposition shows that this is indeed the case.

Proposition 1.4

Let $\beta \in \mathbb {R}{\setminus }\{0\}$ and fix $(u_0, u_1) \in \mathcal {H}^s(\mathbb {T}^2)$ for some $s > 0$. Given any small $T>0$, the solution $u_N$ to (1.21) converges in probability to the solution u, satisfying the following linear stochastic wave equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u + (1-\Delta ) u = \xi \\ (u,\partial _tu)|_{t=0}=(u_0,u_1) \end{array}\right. } \end{aligned}$$

(1.22)

in the class $C([0,T];H^{-\varepsilon }(\mathbb {T}^2))$, $\varepsilon > 0$, as $N \rightarrow \infty$.

We point out that the nature of “triviality” in Proposition 1.4 is slightly different from that in the work [23, 33, 35] with the cubic nonlinearity, where solutions $u_N$ to the unrenormalized equation with regularized noises tend to 0 as we remove the regularization. This difference comes from the different renormalization procedures; our renormalization for the sine-Gordon model appears as a multiplication by the renormalization constant. On the other hand, in the case of the cubic nonlinearity, the renormalization is implemented by insertion of a linear term $\gamma _N u_N$ with a suitable divergent renormalization constant $\gamma _N$, thus modifying the linear part of the equation. In particular, in the case of the wave equation the modified linear propagator introduces faster and faster oscillations and thus the contribution from the deterministic initial data $(u_0, u_1)$ also vanishes. See [33, 35].

Following the previous work [23, 33, 35], the main idea for proving Proposition 1.4 is to artificially insert the renormalization constant $\gamma _N$ in the unrenormalized equation (1.21). By using the decomposition $u_N = \Psi _N + v_N$ as before, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2v_N + (1-\Delta )v_N + \gamma _N^{-1} {\hbox {Im}}\big (\Theta _N e^{i\beta v_N}\big )=0\\ (v_N,\partial _tv_N)\big |_{t=0}=(u_0,u_1). \end{array}\right. } \end{aligned}$$

Proposition 1.4 then follows from essentially repeating the proof of Theorem 1.2 along with the asymptotic behavior (1.18) of the renormalization constant.

Remark 1.5

In the case of a polynomial nonlinearity, the local-in-time analysis in [19] essentially applies to the following hyperbolic $\Phi ^{2k+2}_2$-model⁸:

$$\begin{aligned} \partial _t^2u + \partial _tu+ (1-\Delta )u + u^{2k+1} =\sqrt{2}\xi \end{aligned}$$

(1.23)

with random initial data distributed by the Gibbs measure ($= \Phi ^{2k+2}_2$-measure). Then, Bourgain’s invariant measure argument [6, 7] allows us to establish almost sure global well-posedness of (a renormalized version of) (1.23) and invariance of the Gibbs measure. See [21, 37].

By adding the damping term $\partial _tu$ to (1.1), we obtain the following “canonical” stochastic sine-Gordon model:

$$\begin{aligned} \partial _t^2u+ \partial _tu + (1-\Delta )u + \gamma \sin (\beta u) = \sqrt{2}\xi . \end{aligned}$$

(1.24)

This equation is a hyperbolic counterpart of the stochastic quantization equation for the quantum sine-Gordon model; the Eq. (1.24) formally preserves the Gibbs measure $\mu$ given by⁹

$$\begin{aligned} d\mu {=} Z^{-1} \exp \bigg ( {-} \frac{1}{2} \int _{\mathbb {T}^2} |\langle \nabla \rangle u|^2 dx{-} \frac{1}{2} \int _{\mathbb {T}^2} (\partial _tu)^2 dx {+} \frac{\gamma }{\beta }\int _{\mathbb {T}^2} \cos (\beta u) dx\bigg ) du \otimes d (\partial _tu). \end{aligned}$$

Hence, it is of great importance to study the well-posedness issue of (1.24). For this model, by applying an analogous renormalization and repeating the analysis in Sect. 2 below, we see that the renormalization constant is time-independent (as in the parabolic model [8, 24]). In particular, the regularity depicted in Proposition 1.1 for the relevant imaginary Gaussian chaos $\Theta$ will be time-independent in this case, implying that there exists an infinite number of threshold values for $\beta ^2$ as in the parabolic case. By drawing an analogy to the $\Phi ^3_d$-model (see [24]), we see that the $\Phi ^3_3$-model corresponds to $\beta ^2 = 2\pi$. In a recent preprint [36], we establish almost sure global well-posedness of (a renormalized version of) (1.24) and invariance of the Gibbs measure for $0< \beta ^2 < 2\pi$.

In view of a recent work [20], as for the stochastic nonlinear wave equation on $\mathbb {T}^3$ with a quadratic nonlinearity, we expect that it would require a significant effort (such as establishing multilinear smoothing and introducing appropriate paracontrolled operators) to reach the first non-trivial threshold $\beta ^2 = 2\pi$ for the hyperbolic sine-Gordon model (1.24).¹⁰

Remark 1.6

Albeverio et al. [1] considered the following stochastic nonlinear wave equation:

$$\begin{aligned} \partial _t^2u +(1-\Delta ) u + f(u)=\xi \end{aligned}$$

(1.25)

with zero initial data, where f is assumed to be smooth and bounded. Working within the framework of Colombeau generalized functions, they showed that the solutions $u_N$ to (1.25) with regularized noises tend to the solution u to the linear stochastic wave equation (1.22). It is, however, not clear what the meaning of the solutions constructed in [1] and how they relate to the construction in this paper. Note that in our triviality result (Proposition 1.4), we establish the convergence in the natural space $C([0, T]; H^{-\varepsilon }(\mathbb {T}^2))$. See also the comments in [8, 24] on the work [2] in the parabolic setting.

Notations Before proceeding further, we introduce some notations here. We set

$$\begin{aligned} e_n(x) {\mathop {=}\limits ^{def }} \frac{1}{2\pi }e^{in\cdot x}, \quad n\in \mathbb {Z}^2, \end{aligned}$$

(1.26)

for the orthonormal Fourier basis in $L^2(\mathbb {T}^2)$. Given $s \in \mathbb {R}$, we define the Bessel potential $\langle \nabla \rangle ^s$ of order $-s$ as a Fourier multiplier operator given by

$$\begin{aligned} \langle \nabla \rangle ^s f = \mathcal {F}^{-1} (\langle n \rangle ^s \widehat{f} (n)), \end{aligned}$$

where $\mathcal {F}^{-1}$ denotes the inverse Fourier transform and $\langle \,\cdot \, \rangle = (1+|\cdot |^2)^\frac{1}{2}$. Then, we define the Sobolev space $H^s (\mathbb {T}^2)$ by the norm:

$$\begin{aligned} \Vert f\Vert _{H^s(\mathbb {T}^2)} = \Vert \langle \nabla \rangle ^s f \Vert _{L^2} = \Vert \langle n \rangle ^s\widehat{f}(n)\Vert _{\ell ^2(\mathbb {Z}^2)}. \end{aligned}$$

We also set

$$\begin{aligned} \mathcal {H}^{s}(\mathbb {T}^2) {\mathop {=}\limits ^{def }} H^{s}(\mathbb {T}^2) \times H^{s-1}(\mathbb {T}^2). \end{aligned}$$

We use short-hand notations such as $C_TH^s_x = C([0, T]; H^s(\mathbb {T}^2))$ and $L^p_\omega = L^p(\Omega )$.

For $A, B > 0$, we use $A\lesssim B$ to mean that there exists $C>0$ such that $A \le CB$. By $A\sim B$, we mean that $A\lesssim B$ and $B \lesssim A$. We also use a subscript to denote dependence on an external parameter; for example, $A\lesssim _{\alpha } B$ means $A\le C(\alpha ) B$, where the constant $C(\alpha ) > 0$ depends on a parameter $\alpha$. Given two functions f and g on $\mathbb {T}^2$, we write

$$\begin{aligned} f\approx g \end{aligned}$$

(1.27)

if there exist some constants $c_1,c_2\in \mathbb {R}$ such that $f(x)+c_1 \le g(x) \le f(x)+c_2$ for any $x\in \mathbb {T}^2\backslash \{0\} \cong [-\pi , \pi )^2 {\setminus }\{0\}$. Given $A, B \ge 0$, we also set $A\vee B = \max (A, B)$ and $A\wedge B = \min (A, B)$.

2 On the imaginary Gaussian multiplicative chaos

In this section, we study the regularity and convergence properties of the imaginary complex Gaussian chaos $\Theta _N = \, :\!e^{i\beta \Psi _N}\!:$ defined in (1.16) and present a proof of Proposition 1.1. As pointed out in the introduction, the main difficulty arises in that the processes $\Theta _N$ do not belong to any Wiener chaos of finite order. This forces us to estimate all the higher moments by hand. As in [24], the main ingredient for proving Proposition 1.1 is the bound (2.15) in Lemma 2.5, exhibiting a certain cancellation property. This allows us to simplify the expressions of the moments.

2.1 Preliminary results

We first recall the Poisson summation formula (see [17, Theorem 3.2.8]). In our context, it reads as follows. Let $f \in L^1(\mathbb {R}^d)$ such that (i) $|f(x)| \le \langle x \rangle ^{-d - \delta }$ for some $\delta > 0$ and any $x \in \mathbb {R}^d$ and (ii) $\sum _{n \in \mathbb {Z}^d} |\widehat{f}(n)| < \infty$, where $\widehat{f}$ denotes the Fourier transform of f on $\mathbb {R}^d$ defined by

$$\begin{aligned} \widehat{f}(\xi ) = \frac{1}{(2\pi )^\frac{d}{2}} \int _{\mathbb {R}^d} f(x) e^{-i\xi \cdot x} d x. \end{aligned}$$

Then, we have

$$\begin{aligned} \sum _{n\in \mathbb {Z}^d}\widehat{f}(n)e_n(x) = \sum _{m\in \mathbb {Z}^d}f(x+2\pi m) \end{aligned}$$

(2.1)

for any $x \in \mathbb {R}^d$.

We also recall the following calculus lemma from [23, Lemma 3.2].

Lemma 2.1

There exists $C>0$ such that

$$\begin{aligned} \left| \sum _{\begin{array}{c} n \in \mathbb {Z}^2\\ |n|\le R \end{array}}\frac{1}{a+|n|^2}-\pi \log \left( 1+\frac{R^2}{a}\right) \right| \le \frac{C}{\sqrt{a}}\min \left( 1,\frac{R}{\sqrt{a}}\right) \end{aligned}$$

for any $a,R\ge 1$.

In [19, 20, 39], the analysis of stochastic objects was carried out on the Fourier side. It turns out that for our problem, it is more convenient to work on the physical side. For this reason, we recall several facts from harmonic analysis. For $d\in \mathbb {N}$ and $\alpha >0$, we denote by $\langle \nabla \rangle ^{-\alpha } = (1-\Delta )^{-\frac{\alpha }{2}}$ the Bessel potential of order $\alpha$. This operator on $\mathbb {T}^d$ is given by a convolution with the following distribution:

$$\begin{aligned} J_{\alpha }(x) {\mathop {=}\limits ^{\text {def}}}\lim _{N \rightarrow \infty }\frac{1}{2\pi } \sum _{n \in \mathbb {Z}^d} \frac{\chi _N(n)}{\langle n \rangle ^{\alpha }} e_n(x), \end{aligned}$$

(2.2)

where $\chi _N$ is the smooth cutoff function onto the frequencies $\{|n|\lesssim N\}$ defined in (1.7). It is then known that for $0<\alpha <d$, the distribution $J_\alpha$ behaves like $|x|^{-d + \alpha }$ modulo an additive smooth function on $\mathbb {T}^d$. More precisely, we have the following lemma.

Lemma 2.2

Let $0< \alpha < d$. Then, the function $J_\alpha$ in (2.2) is smooth away from the origin and integrable on $\mathbb {T}^d$. Furthermore, there exist a constant $c_{\alpha ,d}$ and a smooth function R on $\mathbb {T}^d$ such that

$$\begin{aligned} J_\alpha (x) = c_{\alpha ,d} |x|^{\alpha - d} +R(x) \end{aligned}$$

for all $x\in \mathbb {T}^d{\setminus }\{0\} \cong [-\pi , \pi )^d {\setminus }\{0\}$.

Proof

Let $K_{\alpha }$ be the convolution kernel for the Bessel potential on $\mathbb {R}^d$. Then, it follows from [18, Proposition 1.2.5] that $K_\alpha$ is a smooth function on $\mathbb {R}^d{\setminus }\{0\}$ and decays exponentially as $|x|\rightarrow \infty$. Moreover, the following asymptotic behavior holds:

$$\begin{aligned} K_\alpha (x) = c_{\alpha ,d} |x|^{\alpha - d} + o(1) \end{aligned}$$

as $x \rightarrow 0$. See (4,2) in [3]. Then, this lemma follows from applying the Poisson summation formula (2.1) with a frequency truncation $\mathbf {P}_N$ and then taking $N \rightarrow \infty$ as in [17, Example 3.2.10]. See also [5]. $\square$

Next, we study the Green’s function for $1- \Delta$. Recall from [18, Proposition 1.2.5] that the Green’s function $G_{\mathbb {R}^2}$ for $1 - \Delta$ on $\mathbb {R}^2$ is a smooth function on $\mathbb {R}^2 {\setminus } \{0\}$ and decays exponentially as $|x| \rightarrow \infty$. Furthermore, it satisfies

$$\begin{aligned} G_{\mathbb {R}^2}(x) = -\frac{1}{2\pi } \log |x| + o(1) \end{aligned}$$

(2.3)

as $x \rightarrow 0$. See (4,2) in [3]. Now, let G be the Green’s function for $1-\Delta$ on $\mathbb {T}^2$. In view of our normalization (1.26), we then have

$$\begin{aligned} G {\mathop {=}\limits ^{\text {def}}}(1-\Delta )^{-1} \delta _0 = \frac{1}{2\pi }\sum _{n\in \mathbb {Z}^2}\frac{1}{\langle n \rangle ^{2}}e_n, \end{aligned}$$

(2.4)

where the sum is to be interpreted in the limiting sense as in (2.2). Then, by applying the Poisson summation formula (2.1) (with a frequency truncation $\mathbf {P}_N$ and taking $N \rightarrow \infty$) together with the asymptotics (2.3), we obtain

$$\begin{aligned} G(x) = -\frac{1}{2\pi } \log |x| + R(x), \quad x \in \mathbb {T}^2 {\setminus } \{0\}, \end{aligned}$$

(2.5)

for some smooth function R.

In the next lemma, we establish an analogous behavior for the frequency-truncated Green’s function $\mathbf {P}_N^2 G$ given by

$$\begin{aligned} \mathbf {P}_N^2G(x) = \frac{1}{2\pi }\sum _{n\in \mathbb {Z}^2}\frac{\chi _N^2(n)}{\langle n \rangle ^2}e_n(x). \end{aligned}$$

(2.6)

See also Lemma 3.7 in [24].

Lemma 2.3

For any $N\in \mathbb {N}$ and $x\in \mathbb {T}^2{\setminus } \{0\}$, we have

$$\begin{aligned} \mathbf {P}_N^2G(x) \approx -\frac{1}{2\pi } \log \big (|x|+N^{-1}\big ), \end{aligned}$$

where the notation $\approx$ is as in (1.27).

Proof

Fix $x\in \mathbb {T}^2 {\setminus }\{0\}\cong [-\pi ,\pi )^2{\setminus }\{0\}$. We separately consider the cases $|x|\lesssim N^{-1}$ and $|x| > rsim N^{-1}$.

$\bullet$ Case 1 We first consider the case $|x|\lesssim N^{-1}$. By the mean value theorem, we have

$$\begin{aligned} \big |\mathbf {P}_N^2G(x) - \mathbf {P}_N^2G(0)\big | \sim \bigg |\sum _{n\in \mathbb {Z}^2}\frac{\chi _N^2(n)}{\langle n \rangle ^{2}}(e_n(x)-e_n(0))\bigg | \lesssim \sum _{|n|\lesssim N} \frac{|x|}{\langle n \rangle } \lesssim N |x| \lesssim 1. \end{aligned}$$

(2.7)

On the other hand, from Lemma 2.1, we have

$$\begin{aligned} \mathbf {P}_N^2G(0) = \frac{1}{4\pi ^2}\sum _{n\in \mathbb {Z}^2}\frac{\chi _N^2(n)}{\langle n \rangle ^2}=\frac{1}{2\pi }\log N + O(1) \end{aligned}$$

(2.8)

as $N\rightarrow \infty$. Hence, from (2.7) and (2.8), we obtain

$$\begin{aligned} \mathbf {P}_N^2G (x) \approx -\frac{1}{2\pi }\log N^{-1} \approx -\frac{1}{2\pi }\log \big (|x| + N^{-1}\big ) \end{aligned}$$

in this case.

$\bullet$ Case 2 Next, we consider the case $|x| > rsim N^{-1}$. Let $\rho \in C^{\infty }(\mathbb {R}^2)$ with a compact support in $\mathbb {T}^2\cong [-\pi ,\pi )^2$ with $\int \rho (y)dy = 1$. Given $M \ge N$, let $\rho _{M}$ denote a mollification kernel of scale $M^{-1}$ given by $\rho _{M}(x)=M^2\rho (Mx)$. Then, since G and $\mathbf {P}_N^2G$ are smooth away from the origin, we have

$$\begin{aligned} \mathbf {P}_N^2G(x)-G(x)=\lim _{M\rightarrow \infty }\rho _{M}*\big [(\mathbf {P}_N^2-1)G\big ](x). \end{aligned}$$

(2.9)

Then, by the Poisson summation formula (2.1), we write

$$\begin{aligned} \begin{aligned} \rho _{M}*\big [(\mathbf {P}_N^2-1)G\big ](x)&= \frac{1}{2\pi }\sum _{n\in \mathbb {Z}^2}\widehat{\rho }_{M}(n) \frac{\chi _N^2(n)-1}{\langle n \rangle ^{2}}e_n(x)\\&=\frac{1}{2\pi } \sum _{m\in \mathbb {Z}^2}f_{N,M}(x+2\pi m), \end{aligned} \end{aligned}$$

(2.10)

where $f_{N, M} \in \mathcal {S}(\mathbb {R}^2)$ is defined by its Fourier transform:

$$\begin{aligned} \widehat{f}_{N,M}(\xi ) = \widehat{\rho }_{M}(\xi )(\chi _N^2(\xi )-1)\langle \xi \rangle ^{-2}\in \mathcal {S}(\mathbb {R}^2). \end{aligned}$$

By applying integrations by parts with the properties of $\chi _N$ and $\rho _{M}$, in particular, the fact that the integrand is essentially¹¹ supported on $\{N \lesssim |\xi |\lesssim M\}$, we obtain

$$\begin{aligned} \begin{aligned} \big |f_{N,M}(x+2\pi m)\big |&= \frac{1}{2\pi } \bigg |\int _{\mathbb {R}^2}\widehat{f}_{N,M}(\xi )e^{i\xi \cdot (x+2\pi m)}d\xi \bigg |\\&\sim |x{+}2\pi m|^{-2k} \bigg |\int _{\mathbb {R}^2}\Delta _\xi ^k\big (\widehat{\rho }_{M}(\xi )(\chi _N^2(\xi ){-}1) \langle \xi \rangle ^{-2}\big )e^{ i\xi \cdot (x+2\pi m)}d\xi \bigg |\\&\lesssim N^{-2k}|x+2\pi m|^{-2k} \end{aligned} \end{aligned}$$

(2.11)

for any $k\in \mathbb {N}$, uniformly in $M\ge N \ge 1$. When a derivative hits the second factor $\chi _N^2(\xi )-1$, we used the fact that the support in that case is essentially contained in $\{|\xi |\sim N\}$. When no derivative hits the second factor $\chi _N^2(\xi )-1$ or the third factor $\langle \xi \rangle ^{-2}$, namely, when all the derivatives hit the first factor $\widehat{\rho }_{M}(\xi )=\widehat{\rho }(M^{-1}\xi )$, we used the following bound:

$$\begin{aligned} M^{-2k} \log \frac{M}{N} = N^{-2k} \bigg (\frac{N}{M}\bigg )^{2k} \log \frac{M}{N} \lesssim N^{-2k}. \end{aligned}$$

By choosing $k > 1$, we perform the summation over $m \in \mathbb {Z}^2$ and take the limit $M\rightarrow \infty$ in (2.10). From (2.9), (2.10), and (2.11), we obtain

$$\begin{aligned} \begin{aligned} |\mathbf {P}_N^2G(x)-G(x)|&\lesssim N^{-2k}|x|^{-2k} + \sum _{m\ne 0}N^{-2k}|m|^{-2k} \lesssim N^{-2k}(|x|^{-2k}+1)\\&\lesssim (N|x|)^{-2k} \lesssim 1, \end{aligned} \end{aligned}$$

(2.12)

under our assumption $|x| > rsim N^{-1}$, where we used that $|x+2\pi m| > rsim |m|$ for $x\in \mathbb {T}^2\cong [-\pi ,\pi )^2$ and $m\ne 0$. Finally, from (2.5) and (2.12), we obtain

$$\begin{aligned} \mathbf {P}_N^2G (x)\approx G(x) \approx -\frac{1}{2\pi }\log |x| \approx -\frac{1}{2\pi }\log \big (|x|+N^{-1}\big ). \end{aligned}$$

This completes the proof of Lemma 2.3. $\square$

Remark 2.4

Let $N_2 \ge N_1 \ge 1$. Then, by slightly modifying the proof of Lemma 2.3 (namely, replacing $\chi _N^2$ by $\chi _{N_1} \chi _{N_2}$), we have

$$\begin{aligned} \mathbf {P}_{N_1}\mathbf {P}_{N_2} G(x) \approx -\frac{1}{2\pi } \log \big (|x|+N_1^{-1}\big ). \end{aligned}$$

(2.13)

Similarly, we have

$$\begin{aligned} |\mathbf {P}_{N_j}^2 G(x) - \mathbf {P}_{N_1}\mathbf {P}_{N_2} G(x)| \lesssim \big (1 \vee - \log \big (|x|+N_2^{-1}\big )\big ) \wedge \big (N_1^{-1}|x|^{-1} \big ) \end{aligned}$$

(2.14)

for $j = 1, 2$. We point out that the second bound by $N_1^{-1}|x|^{-1}$ in (2.14) follows from the computation up to (2.12) in Case 2 of the proof of Lemma 2.3 (without dividing the argument, depending on the size of |x|).

The next lemma plays a crucial role in proving Proposition 1.1, allowing us to reduce the product of $p(2p-1)$ factors to that of p factors.

Lemma 2.5

Let $\lambda >0$ and $p \in \mathbb {N}$. Given $j \in \{1, \ldots , 2p\}$, we set $\varepsilon _j = 1$ if j is even and $\varepsilon _j = -1$ if j is odd. Let $S_p$ be the set of all permutations of the set $\{1,2,\ldots , p\}$. Then, the following estimate holds:

$$\begin{aligned} \prod _{1\le j< k \le 2p} \big (|y_j - y_k| + N^{-1}\big )^{ \varepsilon _j\varepsilon _k\lambda } \lesssim \max _{\tau \in S_p} \prod _{1\le j \le p} \big (|y_{2j} - y_{2\tau (j)-1}| + N^{-1}\big )^{ -\lambda } \end{aligned}$$

(2.15)

for any set $\{y_j\}_{j=1,\ldots ,2p}$ of 2p points in $\mathbb {T}^2$ and any $N \in \mathbb {N}$.

In [24], Hairer and Shen established an analogous result for a more general class of “potential” function $\mathcal {J}_N(x-y)$ than $(|x-y|+N^{-1})^{\lambda }$ considered above. They exploited sophisticated multi-scale analysis and charge cancellation. In particular, Lemma 2.5 follows from Proposition 3.5 in [24]. Nonetheless, we decided to include a proof since the relevant computation is elementary and concise in our concrete setting. In the proof below, the charge cancellation due to the “dipole” formation appears in (2.19) and (2.20). We also mention the work [13] where an exact identity (see (3.13) in [13]) was used to study a similar problem.

Proof

We prove (2.15) by induction on p. When $p = 1$, (2.15) holds with an equality. Now, let us assume that (2.15) holds for some $p\in \mathbb {N}$ and fix a set $\{y_j\}_{j=1,\ldots ,2(p+1)}$ of $2(p+1)$ points in $\mathbb {T}^2$. By defining the sets $A_p^+ = \{j=1,\ldots ,2p,~j\text { even}\}$ and $A_p^- = \{k=1,\ldots ,2p, ~k\text { odd}\}$, we set

$$\begin{aligned} A_p^{\sigma _1 \sigma _2} = \big \{ (j, k) \in \{1, \ldots , 2p\}: j < k, \ j \in A_p^{\sigma _1}, \ k \in A_p^{\sigma _2}\big \} \end{aligned}$$

for $\sigma _1, \sigma _2\in \{ +, -\}$. With a slight abuse of notation, we identify $+$ and − with $+1$ and $-1$ in the following.

We denote by $\Pi _{p}$ the left-hand side of (2.15). Then, we have

$$\begin{aligned} \Pi _{p+1} = \frac{\prod _{\begin{array}{c} \sigma _1, \sigma _2 \in \{+, -\}\\ \sigma _1\sigma _2 = 1 \end{array}}\prod _{(j,k)\in A_{p+1}^{\sigma _1\sigma _2}} \big (|y_j - y_k| + N^{-1}\big )^{ \lambda } }{\prod _{\begin{array}{c} \sigma _1, \sigma _2 \in \{+, -\}\\ \sigma _1\sigma _2 = -1 \end{array}}\prod _{(j,k) \in A_{p+1}^{\sigma _1\sigma _2}} \big (|y_j - y_k| + N^{-1}\big )^{ \lambda }} . \end{aligned}$$

Namely, the numerator contains all the factors with the same parity for j and k, while the denominator contains all the factors with the opposite parities for j and k. Now, choose $j_0 \in A_{p+1}^+$ and $k_0 \in A_{p+1}^-$ such that

$$\begin{aligned} |y_{j_0} - y_{k_0}| = \min \big \{ |y_j - y_k|; j \in A_{p+1}^+, k \in A_{p+1}^-\big \}. \end{aligned}$$

(2.16)

It follows from its definition given by the left-hand side of (2.15) (with p replaced by $p+1$) that $\Pi _{p+1}$ is invariant under permutations of $\{1, \ldots , 2p+2\}$ that do not mix even and odd integers (i.e. not mixing the $+$ and − charges). Thus, without loss of generality, we assume that $(j_0,k_0) = (2p+1, 2p+2)$.

By the inductive hypothesis we have

$$\begin{aligned} \Pi _p \lesssim \max _{\tau \in S_p} \prod _{1\le j\le p} \big (|y_{2j} - y_{2\tau (j)-1}| + N^{-1}\big )^{- \lambda }. \end{aligned}$$

In view of (2.15), it suffices to prove the following bound:

$$\begin{aligned} \frac{\Pi _{p+1}}{\Pi _p} \lesssim \big (|y_{2p+1} - y_{2p+2}| + N^{-1}\big )^{- \lambda }. \end{aligned}$$

(2.17)

uniformly in N. Note that the left-hand side of (2.17) contains only the factors involving $y_{2p+1}$ or $y_{2p+2}$. Hence, we have

$$\begin{aligned} \begin{aligned} \frac{\Pi _{p+1}}{\Pi _p} \big (&|y_{2p+1} - y_{2p+2}| + N^{-1}\big )^{\lambda }\\&= \prod _{1\le j \le p} \bigg ( \frac{|y_{2j-1} - y_{2p+1}| + N^{-1}}{|y_{2j-1} - y_{2p+2}| + N^{-1}}\bigg )^{\lambda } \bigg ( \frac{|y_{2j} - y_{2p+2}| + N^{-1}}{|y_{2j} - y_{2p+1}| + N^{-1}}\bigg )^{\lambda }. \end{aligned} \end{aligned}$$

(2.18)

On the other hand, by the triangle inequality and (2.16), we have

$$\begin{aligned} \max _{1\le j \le p} \frac{|y_{2j-1} - y_{2p+1}| + N^{-1}}{|y_{2j-1} - y_{2p+2}| + N^{-1}} \le 1 + \max _{1\le j \le p} \frac{ |y_{2p+1} - y_{2p+2}|}{|y_{2j-1} - y_{2p+2}| + N^{-1}} \lesssim 1, \end{aligned}$$

(2.19)

uniformly in N. Similarly, we have

$$\begin{aligned} \max _{1 \le j \le p} \frac{|y_{2j} - y_{2p+2}| + N^{-1}}{|y_{2j} - y_{2p+1}| + N^{-1}} \lesssim 1, \end{aligned}$$

(2.20)

uniformly in N. Therefore, the desired estimate (2.17) follows from (2.18), (2.19), and (2.20). $\square$

2.2 Estimates on the stochastic objects

In this subsection, we present a proof of Proposition 1.1 on the imaginary complex Gaussian chaos $\Theta _N = \, :\!e^{i\beta \Psi _N}\!:$ defined in (1.16).

We first recall the following lemma from [19, Proposition 2.1] on the regularity of the truncated stochastic convolution $\Psi _N$. See also [20, Lemma 3.1].

Lemma 2.6

Given any $T,\varepsilon >0$ and finite $p \ge 1$, $\{\Psi _N\}_{N\in \mathbb {N}}$ is a Cauchy sequence in $L^p(\Omega ;C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2)))$, converging to some limit $\Psi$ in $L^p(\Omega ;C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2)))$. Moreover, $\Psi _N$ converges almost surely to the same limit $\Psi \in C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2))$.

It is easy to show that the claim in Lemma 2.6 fails when $\varepsilon = 0$. Thus, as $N\rightarrow \infty$, $\Psi _N$ becomes singular, necessitating a proper renormalization procedure for $\Theta _N$ defined in (1.16).

As mentioned above, we will work on the physical space (rather than on the Fourier side as in [19, 20, 39]). For this purpose, we first study the covariance function for $\Psi _N$. Let

$$\begin{aligned} \Gamma _N(t,x-y) {\mathop {=}\limits ^{\text {def}}}\mathbb {E}\big [\Psi _N(t,x) \Psi _N(t,y)\big ]. \end{aligned}$$

(2.21)

Lemma 2.7

Given $N \in \mathbb {N}$, let $\Gamma _N$ be as in (2.21). Then, we have

$$\begin{aligned} \Gamma _N(t,x-y) =\frac{1}{2\pi }\sum _{n\in \mathbb {Z}^2} \chi _N^2(n) \bigg \{ \frac{t}{2 \langle n \rangle ^2} - \frac{\sin (2 t \langle n \rangle )}{4\langle n \rangle ^3 }\bigg \}e_n (x-y) . \end{aligned}$$

(2.22)

In particular, we have

$$\begin{aligned} \Gamma _N(t,x-y) \approx - \frac{t}{4\pi } \log \big (|x-y| + N^{-1}\big ) \end{aligned}$$

(2.23)

for any $t \in \mathbb {R}$.

Proof

The identity (2.22) follows from a straightforward computation, using (1.10) and (1.11) with (1.26). See (1.8) and (2.8) in [19] for analogous computations.

From (2.6) and (2.22), we have

$$\begin{aligned} \Gamma _N (t, x) \approx \frac{t}{2} \mathbf {P}_N^2 G(x). \end{aligned}$$

Then, (2.23) follows from Lemma 2.3. $\square$

By setting

$$\begin{aligned} \Gamma (t,x-y) {\mathop {=}\limits ^{\text {def}}}\mathbb {E}\big [\Psi (t,x) \Psi (t,y)\big ], \end{aligned}$$

we then formally have

$$\begin{aligned} \Gamma (t,x-y) =\frac{1}{2\pi }\sum _{n\in \mathbb {Z}^2} \bigg \{ \frac{t}{2 \langle n \rangle ^2} - \frac{\sin (2 t \langle n \rangle )}{4\langle n \rangle ^3 }\bigg \}e_n (x-y) , \end{aligned}$$

(2.24)

where the sum is to be interpreted in the limiting sense as in (2.2). By comparing (2.4) and (2.24), we have $\Gamma (t, x) \approx \frac{t}{2} G(x)$. Similarly, we have

$$\begin{aligned} \mathbf {P}_{N_1} \mathbf {P}_{N_2} \Gamma (t, x) \approx \frac{t}{2} \mathbf {P}_{N_1}\mathbf {P}_{N_2} G(x). \end{aligned}$$

(2.25)

In particular, from (2.13) in Remark 2.4, we obtain

$$\begin{aligned} \mathbf {P}_{N_1}\mathbf {P}_{N_2} \Gamma (t, x) \approx -\frac{t}{4\pi } \log \big (|x|+N_1^{-1}\big ) \end{aligned}$$

(2.26)

for $N_2 \ge N_1 \ge 1$.

We are now ready to present a proof of Proposition 1.1.

Proof of Proposition 1.1

Fix $\beta \in \mathbb {R}$ and $T > 0$ such that $\beta ^2 T < 8 \pi$. Also let $p \in \mathbb {N}$, finite $q \ge 1$, and $\alpha > \frac{\beta ^2 T}{8\pi }$. Without loss of generality, we assume $\alpha < 2$ in the following.

Fix small $\delta > 0$. Then, by Sobolev’s inequality (in x) followed by Minkowski’s integral inequality, we have

$$\begin{aligned} \Vert \Theta _N\Vert _{L^{2p}_\omega L^q_TW^{-\alpha ,\infty }_x} \lesssim \Vert \langle \nabla \rangle ^{\delta -\alpha }\Theta _N\Vert _{L^{2p}_\omega L^q_TL^{r_{\delta }}_x}\lesssim \Big \Vert \Vert \langle \nabla \rangle ^{\delta -\alpha }\Theta _N(t,x)\Vert _{L^{2p}_\omega }\Big \Vert _{L^q_TL^{r_{\delta }}_x} \end{aligned}$$

(2.27)

for some large but finite $r_{\delta }$, provided that $2p\ge \max (q,r_{\delta })$.

Fix $t\in [0,T]$ and $x\in \mathbb {T}^2$. Recalling $\langle \nabla \rangle ^{\delta - \alpha } f = J_{\alpha -\delta } * f$, where $J_\alpha$ is as in (2.2), it follows from (1.16) that

$$\begin{aligned} \mathbb {E}\Big [\big | \langle \nabla \rangle ^{\delta -\alpha }\Theta _N(t,x)\big |^{2p}\Big ]&= e^{p\beta ^2\sigma _N(t)}\mathbb {E}\Bigg [ \bigg | \int _{\mathbb {T}^2} J_{\alpha -\delta } (x-y) e^{i \beta \Psi _N (t,y)} dy \bigg |^{2p} \Bigg ]\nonumber \\&= e^{p\beta ^2\sigma _N(t)}\int _{(\mathbb {T}^{2})^{2p}} \mathbb {E}\bigg [ e^{i \beta \sum _{j =1}^p (\Psi _N (t,y_{2j}) - \Psi _N (t,y_{2j-1}))} \bigg ]\nonumber \\&\quad \,\,\times \prod _{k=1}^{2p} J_{\alpha -\delta } (x-y_k) d\vec y, \end{aligned}$$

(2.28)

where $d\vec y {\mathop {=}\limits ^{\text {def}}}dy_1\cdots d y_{2p}$. Noting that $\sum _{j =1}^p (\Psi _N (t,y_{2j}) - \Psi _N (t,y_{2j-1}))$ is a mean-zero Gaussian random variable, the explicit formula for the characteristic function of a Gaussian random variable yields

$$\begin{aligned} \begin{aligned} \mathbb {E}\bigg [ e^{i \beta \sum _{j =1}^p (\Psi _N (t,y_{2j}) - \Psi _N (t,y_{2j-1}))} \bigg ]&= e^{- \frac{\beta ^2}{2} \mathbb {E}\big [| \sum _{j =1}^p (\Psi _N (t,y_{2j}) - \Psi _N (t,y_{2j-1})) |^2\big ]} . \end{aligned} \end{aligned}$$

(2.29)

Let $\{\varepsilon _j\}_{j=1,\ldots ,2p}$ be as in Lemma 2.5. Then, we can rewrite the expectation in the exponent on the right-hand side of (2.29) as

$$\begin{aligned} \mathbb {E}\Big [\big |\sum _{j=1}^{2p}\varepsilon _j\Psi _N(t,y_j)\big |^2\Big ] = \sum _{j,k=1}^{2p}\varepsilon _j\varepsilon _k \Gamma _N(t,y_j-y_k), \end{aligned}$$

where $\Gamma _N$ is the covariance function defined in (2.21). From the definition (1.12), we have $\Gamma _N(t,0)=\sigma _N(t)$. Hence, we obtain

$$\begin{aligned} e^{- \frac{\beta ^2}{2} \mathbb {E}\big [| \sum _{j =1}^{p} (\Psi _N (t,y_{2j}) - \Psi _N (y_{2j-1},t))|^2\big ]} = e^{- p \beta ^2 \sigma _N(t)} e^{ - \beta ^2 \sum _{1\le j < k\le 2p} \varepsilon _j\varepsilon _k \Gamma _N(t,y_j-y_k) } . \end{aligned}$$

(2.30)

Then, from (2.29), (2.30), the two-sided bound (2.23) in Lemmas 2.7 and 2.5, we obtain

$$\begin{aligned} \begin{aligned} e^{p\beta ^2\sigma _N(t)} \mathbb {E}\bigg [&e^{i \beta \sum _{j =1}^p (\Psi _N (t,y_{2j}) - \Psi _N (t,y_{2j-1}))} \bigg ]\\&\sim \prod _{1\le j < k \le 2p } \big (|y_j - y_k| + N^{-1}\big )^{\varepsilon _j\varepsilon _k \frac{\beta ^2 t}{4 \pi }}\\&\lesssim \max _{\tau \in S_p} \prod _{1\le j \le p} \big (|y_{2j} - y_{2\tau (j)-1}| + N^{-1}\big )^{ -\frac{\beta ^2 t}{4 \pi }}\\&\le \sum _{\tau \in S_p} \prod _{1\le j \le p} \big (|y_{2j} - y_{2\tau (j)-1}| + N^{-1}\big )^{ -\frac{\beta ^2 t}{4 \pi }}. \end{aligned} \end{aligned}$$

(2.31)

Finally, from (2.28) and (2.31) we obtain

$$\begin{aligned} \begin{aligned} \mathbb {E}&\Big [\big | \langle \nabla \rangle ^{\delta -\alpha }\Theta _N(t,x)\big |^{2p}\Big ]\\&\lesssim \sum _{\tau \in S_p}\int _{(\mathbb {T}^{2})^{2p}} \prod _{1\le j \le p} \big (|y_{2j} - y_{2\tau (j)-1}| + N^{-1}\big )^{ -\frac{\beta ^2 t}{4 \pi }} \prod _{k=1}^{2p} |J_{\alpha -\delta } (x-y_k)| d\vec y. \end{aligned} \end{aligned}$$

(2.32)

In the following, we fix $\tau \in S_p$. Then, it suffices to bound each pair of integrals:

$$\begin{aligned} \int _{\mathbb {T}^2}\int _{\mathbb {T}^2}\big (|y_{j} - y_{k}| + N^{-1}\big )^{ -\frac{\beta ^2 t}{4 \pi }} | J_{\alpha -\delta } (x-y_{j})| |J_{\alpha -\delta } (x-y_{k})| dy_{j}dy_{k}, \end{aligned}$$

for an even integer $j=2,\ldots ,2p$ and $k=2\tau (\tfrac{j}{2})-1$. From Lemma 2.2 with $0< \alpha - \delta < 2$, we can bound this integral by

$$\begin{aligned} \begin{aligned} \int _{\mathbb {T}^2}&\int _{\mathbb {T}^2}\big (|y_{j} - y_{k}|+N^{-1}\big )^{ -\frac{\beta ^2 t}{4 \pi }} |x-y_{j}|^{\alpha -\delta -2} |x-y_{k}|^{\alpha -\delta -2} dy_{j}dy_{k}\\&\le \int _{\mathbb {T}^2}\int _{\mathbb {T}^2}|y_{j} - y_{k}|^{ -\frac{\beta ^2 t}{4 \pi }} |x-y_j|^{\alpha -\delta -2}|x-y_k|^{\alpha -\delta -2} dy_{j}dy_k, \end{aligned} \end{aligned}$$

(2.33)

uniformly in $N\in \mathbb {N}$.

• Case 1 $|y_j-y_k|\sim |x-y_k| > rsim |x-y_j|$.    In this case, we bound the integral in (2.33) by

  $$\begin{aligned} \text {RHS of }(2.33)&\lesssim \int _{\mathbb {T}^2} |x-y_k|^{\alpha -\delta -2-\frac{\beta ^2t}{4\pi }} \int _{|x-y_j|\lesssim |x-y_k|} |x- y_j|^{\alpha -\delta -2} dy_jdy_k\\&\lesssim \int _{\mathbb {T}^2}|x-y_k|^{2(\alpha -\delta ) - 2 -\frac{\beta ^2t}{4\pi }}dy_k \lesssim 1, \end{aligned}$$

  where we used $\frac{\beta ^2t}{8\pi }<\alpha -\delta$. The symmetry allows us to handle the case $|y_j-y_k|\sim |x-y_j| > rsim |x-y_k|$.

• Case 2 $|x-y_j|\sim |x-y_k| > rsim |y_j-y_k|$.    In this case, we bound the integral in (2.33) by

  $$\begin{aligned} \text {RHS of }(2.33)&\lesssim \int _{\mathbb {T}^2}|x- y_k |^{2(\alpha -\delta )-4} \int _{|y_j-y_k|\lesssim |x-y_k|}|y_j-y_k|^{-\frac{\beta ^2t}{4\pi }}dy_jdy_k\\&\lesssim \int _{\mathbb {T}^2}|x-y_k|^{2(\alpha -\delta )-2-\frac{\beta ^2t}{4\pi }}dy_k \lesssim 1 \end{aligned}$$

  since $\frac{\beta ^2t}{8\pi } < \min (\alpha - \delta , 1)$.

Putting together (2.32) and the estimates on (2.33), we obtain

$$\begin{aligned} \mathbb {E}\big [\big | \langle \nabla \rangle ^{\delta -\alpha }\Theta _N(t,x)\big |^{2p}\big ]\lesssim 1, \end{aligned}$$

(2.34)

uniformly in $t\in [0,T]$, $x\in \mathbb {T}^2$, and $N\in \mathbb {N}$. Therefore, we conclude from (2.27) and (2.34) that

$$\begin{aligned} \Vert \Theta _N\Vert _{L^{2p}_\omega L^q_TW^{-\alpha ,\infty }_x}\lesssim T^{\frac{1}{q}}, \end{aligned}$$

uniformly in $N\in \mathbb {N}$.

Next, we establish convergence of $\Theta _N$. Let $N_2 \ge N_1 \ge 1$. By repeating the computation as above when $p=1$ with $\Theta _{N_1}-\Theta _{N_2}$ in place of $\Theta _N$, we have

$$\begin{aligned} \begin{aligned}&\mathbb {E}\Big [ \big | \langle \nabla \rangle ^{\delta -\alpha }\big (\Theta _{N_1}(t,x)-\Theta _{N_2}(t,x)\big ) \big |^{2} \Big ] \\&\quad = \int _{\mathbb {T}^2}\int _{\mathbb {T}^2} J_{\alpha -\delta } (x-y)J_{\alpha -\delta } (x-z) \\&\quad \quad \times \mathbb {E}\bigg [\Big (e^{ \frac{\beta ^2}{2} \sigma _{N_1}(t)}e^{i \beta \Psi _{N_1} (t,y)}-e^{ \frac{\beta ^2}{2} \sigma _{N_2}(t)}e^{i \beta \Psi _{N_2} (t,y)}\Big )\\&\quad \quad \times \Big (e^{ \frac{\beta ^2}{2} \sigma _{N_1}(t)}e^{-i \beta \Psi _{N_1} (t,z)}-e^{ \frac{\beta ^2}{2} \sigma _{N_2}(t)}e^{-i \beta \Psi _{N_2} (t,z)}\Big )\bigg ] dydz \\&\quad = \sum _{j = 1}^2 \int _{\mathbb {T}^2}\int _{\mathbb {T}^2} J_{\alpha -\delta } (x-y)J_{\alpha -\delta } (x-z) \Big (e^{ \beta ^2 \Gamma _{N_j}(t,y-z) }-e^{ \beta ^2 \mathbf {P}_{N_1}\mathbf {P}_{N_2}\Gamma (t,y-z)}\Big )dydz, \end{aligned} \end{aligned}$$

(2.35)

where we used $\mathbf {P}_{N_1}\mathbf {P}_{N_2}\Gamma (t,y-z) = \mathbb {E}\big [\Psi _{N_1}(t,y) \Psi _{N_2} (t,z)\big ]$. By the mean value theorem and the bounds (2.23) in Lemma 2.7 and (2.26), we have

$$\begin{aligned} \begin{aligned}&\Big |e^{ \beta ^2 \Gamma _{N_j}(t,x) }-e^{ \beta ^2 \mathbf {P}_{N_1}\mathbf {P}_{N_2} \Gamma (t,x)}\Big |\\&\quad = \bigg |\big (\mathbf {P}_{N_j}^2\Gamma (t,x) - \mathbf {P}_{N_1}\mathbf {P}_{N_2}\Gamma (t,x)\big )\\&\quad \quad \times \int _0^1\beta ^2 \exp \big [\beta ^2\big (\tau \mathbf {P}_{N_j}^2\Gamma (t,x) +(1-\tau )\mathbf {P}_{N_1} \mathbf {P}_{N_2}\Gamma (t,x)\big )\big ]d\tau \bigg |\\&\quad \lesssim \big (|x|+N_2^{-1}\big )^{-\frac{\beta ^2 t}{4\pi }} \big |\mathbf {P}_{N_j}^2\Gamma (t,x) - \mathbf {P}_{N_1}\mathbf {P}_{N_2}\Gamma (t,x)\big |. \end{aligned} \end{aligned}$$

(2.36)

Given $\varepsilon > 0$, there exists $C_\varepsilon > 0$ such that

$$\begin{aligned} | \log y | \le C_\varepsilon y^{-\varepsilon } \end{aligned}$$

(2.37)

for any $0 < y \lesssim 1$. From (2.14) and (2.25) along with (2.37), we have

$$\begin{aligned} \begin{aligned} \big |\mathbf {P}_{N_j}^2\Gamma (t, x)-\mathbf {P}_{N_1} \mathbf {P}_{N_2}\Gamma (t, x)\big |&\lesssim T \big (|x|+N_2^{-1}\big )^{-\varepsilon }\wedge \big (N_1^{-1}|x|^{-1}\big )\\&\lesssim T N_1^{-\varepsilon } |x|^{-2\varepsilon } \end{aligned} \end{aligned}$$

(2.38)

for any $t \in [0, T]$ and non-zero $x\in \mathbb {T}^2 \cong [-\pi ,\pi )^2$. Hence, from (2.35), (2.36), and (2.38) along with Lemma 2.2 (with $0< \alpha - \delta < 2$), we obtain

$$\begin{aligned}&\mathbb {E}\Big [ \big | \langle \nabla \rangle ^{\delta -\alpha }\big (\Theta _{N_1}(t,x)-\Theta _{N_2}(t,x)\big ) \big |^{2} \Big ]\\&\quad \lesssim T \int _{\mathbb {T}^2}\int _{\mathbb {T}^2} |x-y|^{\alpha -\delta -2}|x-z|^{\alpha -\delta -2}N_1^{-\varepsilon }|y-z|^{-2\varepsilon -\frac{\beta ^2t}{4\pi }}dydz. \end{aligned}$$

By taking $\varepsilon > 0$ sufficiently small such that $\frac{\beta ^2t}{8\pi } +\varepsilon < \min (\alpha - \delta , 1)$, we can proceed as in Cases 1 and 2 above and estimate the integrals above by

$$\begin{aligned} \mathbb {E}\Big [ \big | \langle \nabla \rangle ^{\delta -\alpha }\big (\Theta _{N_1}(t,x)-\Theta _{N_2}(t,x)\big ) \big |^{2}\Big ] \lesssim T N_1^{-\varepsilon } \end{aligned}$$

(2.39)

for any $N_2\ge N_1 \ge 1$, $t\in [0,T]$, and $x\in \mathbb {T}^2$.

Given $p \ge 1$, we can interpolate the previous bound (2.39) with (2.34) and obtain

$$\begin{aligned} \mathbb {E}\Big [ \big | \langle \nabla \rangle ^{\delta -\alpha }\big (\Theta _{N_1}(t,x)-\Theta _{N_2}(t,x)\big ) \big |^{p}\Big ] \le C(T) N_1^{-\varepsilon } \end{aligned}$$

for any $N_2\ge N_1 \ge 1$, $t\in [0,T]$, and $x\in \mathbb {T}^2$. Therefore, we conclude that

$$\begin{aligned} \Vert \Theta _{N_1}-\Theta _{N_2}\Vert _{L^{p}_\omega L^q_TW^{-\alpha ,\infty }_x} \le C(T)N_1^{-\tfrac{\varepsilon }{p}}. \end{aligned}$$

This shows that $\Theta _N$ is a Cauchy sequence in $L^p(\Omega ;L^q([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2)))$. This completes the proof of Proposition 1.1. $\square$

3 Proofs of the main results

In this section, we present a proof of local well-posedness of the renormalized stochastic sine-Gordon equation (Theorem 1.2). Given the regularity and convergence properties for the imaginary Gaussian multiplicative chaos $\Theta _N$ (Proposition 1.1), Theorem 1.2 follows from a standard application of the Strichartz estimate (see (3.3) below) as in the work [19] on the stochastic nonlinear wave equation with a polynomial nonlinearity. For this purpose, we first go over the Strichartz estimate along with other useful lemmas in Sect. 3.1. We then present a proof of Theorem 1.2 in Sect. 3.2. We conclude this paper by establishing a triviality result (Proposition 1.4) for the (unrenormalized) stochastic sine-Gordon equation (1.1) in Sect. 3.3.

3.1 Strichartz estimates and other useful lemmas

We begin by recalling the Strichartz estimate for the linear wave equation on $\mathbb {T}^2$. Given $0<s<1$, we say that a pair (q, r) of exponents (and a pair $(\widetilde{q},\widetilde{r})$, respectively) is s-admissible (and dual s-admissible, respectively), if $1\le \widetilde{q}\le 2\le q\le \infty$ and $1<\widetilde{r}\le 2 \le r<\infty$ and if they satisfy the following scaling and admissibility conditions:

$$\begin{aligned} \frac{1}{q} + \frac{2}{r} = 1-s = \frac{1}{\widetilde{q}}+\frac{2}{\widetilde{r}} -2, \quad \frac{2}{q}+\frac{1}{r}\le \frac{1}{2}, \quad \text {and}\quad \frac{2}{\widetilde{q}}+\frac{1}{\widetilde{r}}\ge \frac{5}{2}. \end{aligned}$$

Then, it follows from Lemma 3.3 in [19], which studies the maximization problem for $J_s = \min (\frac{r}{\widetilde{r}}, \frac{q}{\widetilde{q}})$, $0< s < 1$, that given $0< s < 1$, there exist an s-admissible pair $(q,r) = (q(s), r(s))$ and a dual s-admissible pair $(\widetilde{q},\widetilde{r}) = (\widetilde{q}(s),\widetilde{r}(s))$ such that

$$\begin{aligned} q>2\widetilde{q} \quad \text {and}\quad r>2\widetilde{r} . \end{aligned}$$

(3.1)

Let $0< s < 1$. In the remaining part of this paper, we fix pairs (q, r) and $(\widetilde{q},\widetilde{r})$ as above, satisfying (3.1). Given $0<T\le 1$, we then define the Strichartz space:

$$\begin{aligned} X^s(T)=C([0,T];H^s(\mathbb {T}^2))\cap C^1([0,T];H^{s-1}(\mathbb {T}^2))\cap L^q([0,T];L^r(\mathbb {T}^2)) \end{aligned}$$

(3.2)

and its “dual” space:

$$\begin{aligned} N^s(T) = L^1([0,T];H^{s-1}(\mathbb {T}^2)) + L^{\widetilde{q}}([0,T];L^{\widetilde{r}}(\mathbb {T}^2)), \end{aligned}$$

where $X^s(T)$- and $N^s(T)$-norms are given by

$$\begin{aligned} \Vert u \Vert _{X^s(T)} = \Vert u\Vert _{C_TH^s_x} + \Vert \partial _tu \Vert _{C_TH^{s-1}_x} + \Vert u\Vert _{L^q_TL^r_x} \end{aligned}$$

and

$$\begin{aligned} \Vert u \Vert _{N^s(T)} = \inf \Big \{ \Vert u_1\Vert _{L^1_TH^{s-1}_x} + \Vert u_2\Vert _{L^{\widetilde{q}}_TL^{\widetilde{r}}_x}: u = u_1 + u_2 \Big \}. \end{aligned}$$

Then, the solution u to the following linear wave equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2u+(1-\Delta )u = f\\ (u,\partial _tu)\big |_{t=0}=(u_0,u_1) \end{array}\right. } \end{aligned}$$

on [0, T] satisfies the following Strichartz estimate:

$$\begin{aligned} \Vert u\Vert _{X^s(T)} \lesssim \Vert (u_0,u_1)\Vert _{\mathcal {H}^s} + \Vert f\Vert _{N^s(T)}, \end{aligned}$$

(3.3)

uniformly in $T \in [0, 1]$. The Strichartz estimates on $\mathbb {R}^d$ have been studied extensively by many mathematicians. See [16, 26, 30] in the context of the wave equation. For the Klein–Gordon equation under consideration, see [27]. Thanks to the finite speed of propagation, the estimate (3.3) on $\mathbb {T}^2$ follows from the corresponding estimate on $\mathbb {R}^2$.

Next, we recall the following product estimates. See Lemma 3.4 in [19].

Lemma 3.1

Let $0\le s \le 1$.

• (i) Suppose that $1<p_j,q_j,r < \infty$, $\frac{1}{p_j} + \frac{1}{q_j}= \frac{1}{r}$, $j = 1, 2$. Then, we have

  $$\begin{aligned} \Vert \langle \nabla \rangle ^s (fg) \Vert _{L^r(\mathbb {T}^d)} \lesssim \Big ( \Vert f \Vert _{L^{p_1}(\mathbb {T}^d)} \Vert \langle \nabla \rangle ^s g \Vert _{L^{q_1}(\mathbb {T}^d)} + \Vert \langle \nabla \rangle ^s f \Vert _{L^{p_2}(\mathbb {T}^d)} \Vert g \Vert _{L^{q_2}(\mathbb {T}^d)}\Big ). \end{aligned}$$

  (3.4)

• (ii) Suppose that $1<p,q,r < \infty$ satisfy the scaling condition: $\frac{1}{p}+\frac{1}{q}=\frac{1}{r} + \frac{s}{d}$. Then, we have

  $$\begin{aligned} \big \Vert \langle \nabla \rangle ^{-s} (fg) \big \Vert _{L^r(\mathbb {T}^d)} \lesssim \big \Vert \langle \nabla \rangle ^{-s} f \big \Vert _{L^p(\mathbb {T}^d) } \big \Vert \langle \nabla \rangle ^s g \big \Vert _{L^q(\mathbb {T}^d)}. \end{aligned}$$

  (3.5)

Lastly, we recall the fractional chain rule from [15].

Lemma 3.2

Suppose that F is a Lipschitz function with Lipschitz constant L. Then, given any $0< s < 1$ and $1< p < \infty$, we have

$$\begin{aligned} \big \Vert |\nabla |^s F(u)\big \Vert _{L^p(\mathbb {T}^d)} \lesssim L \big \Vert |\nabla |^s u\big \Vert _{L^{p}(\mathbb {T}^d)} \end{aligned}$$

(3.6)

for any $u\in C^{\infty }(\mathbb {T}^d)$.

The fractional chain rule on $\mathbb {R}^d$ was essentially proved in [9].¹² As for the estimate (3.6) on $\mathbb {T}^d$, see [15].

3.2 Local well-posedness

In this subsection, we present a proof of Theorem 1.2. For simplicity, we assume that $0< s < 1$, in which case we have $\sigma = s$ in the statement of Theorem 1.2. Fix $\beta \in \mathbb {R}{\setminus } \{0\}$ and $(u_0,u_1)\in \mathcal {H}^s(\mathbb {T}^2)$. The main idea is to apply the Da Prato–Debussche trick. Namely, we first write the solution $u_N$ to (1.17) as $u_N=\Psi _N + v_N$, where $\Psi _N$ is the truncated stochastic convolution defined in (1.10). Then, the residual term $v_N$ satisfies (1.19). By writing (1.19) in the Duhamel formulation, we have

$$\begin{aligned} v_N(t) = \partial _tS(t)u_0 + S(t)u_1 -\int _0^tS(t-t'){\hbox {Im}}\big (\Theta _Ne^{i\beta v_N}\big )(t')dt', \end{aligned}$$

(3.7)

where S(t) and $\Theta _N$ are as in (1.9) and (1.16), respectively. In the following, we use $B_R$ to denote the ball in $X^s(T)$ of radius $R > 0$ centered at the origin.

Let $\Phi ^N(v_N) = \Phi ^N_{(u_0, u_1), \Theta _N}(v_N)$ denote the right-hand side of (3.7). Let $0 < T \le 1$. By the Strichartz estimate (3.3), we have

$$\begin{aligned} \big \Vert \Phi ^N(v_N) \big \Vert _ {X^s(T)} \lesssim \Vert (u_0, u_1) \Vert _{\mathcal {H}^s} + \Big \Vert {\hbox {Im}}\big ( \Theta _N e^{i\beta v_N}\big ) \Big \Vert _{N^s(T)}. \end{aligned}$$

(3.8)

From (3.22) and (3.23) in [19], we have

$$\begin{aligned} \Vert f\Vert _{L^{q_1}_TW^{\alpha ,r_1}_x}\lesssim \Vert f\Vert _{X^s(T)} \quad \text {and}\quad \Vert f\Vert _{N^s(T)} \lesssim \Vert f\Vert _{L^{\widetilde{q}_1}_TW^{-\alpha ,\widetilde{r}_1}_x}, \end{aligned}$$

(3.9)

where $0<\alpha <\min (s,1-s)$ and the pairs $(q_1,r_1)$ and $(\widetilde{q}_1,\widetilde{r}_1)$ are given by

$$\begin{aligned} \frac{1}{ q_1} = \frac{1-\alpha /s}{q} + \frac{\alpha /s}{\infty }, \quad \frac{1}{r_1} = \frac{1-\alpha /s}{r} + \frac{\alpha /s}{2} \end{aligned}$$

(3.10)

and

$$\begin{aligned} \frac{1}{\widetilde{q}_1} = \frac{\alpha /(1-s)}{1} + \frac{1-\alpha /(1-s)}{\widetilde{q}}, \quad \frac{1}{\widetilde{r}_1} = \frac{\alpha /(1-s)}{2} + \frac{1-\alpha /(1-s)}{\widetilde{r}} . \end{aligned}$$

(3.11)

By applying (3.9), the product estimate (3.5) in Lemma 3.1 (with sufficiently small $\alpha > 0$), and Lemma 3.2, there exists $\theta > 0$ such that

$$\begin{aligned} \begin{aligned} \Big \Vert {\hbox {Im}}\big ( \Theta _N e^{i\beta v_N}\big ) \Big \Vert _{N^s(T)}&\lesssim \Big \Vert {\hbox {Im}}\big ( \Theta _N e^{i\beta v_N}\big ) \Big \Vert _{L^{\widetilde{q}_1}_T W^{-\alpha , \widetilde{r}_1}_x}\\&\lesssim T^\theta \big \Vert \Theta _N \big \Vert _{L^{q_2}_T W^{-\alpha ,\infty }_x} \Vert e^{i\beta v_N} \Vert _{L^{\widetilde{q}_2}_T W^{\alpha ,\widetilde{r}_2}_x} \\&\lesssim T^\theta \big \Vert \Theta _N \big \Vert _{L^{q_2}_T W^{-\alpha ,\infty }_x} \Big (1+\beta \Vert v_N \Vert _{L^{\widetilde{q}_2}_T W^{\alpha ,\widetilde{r}_2}_x}\Big ) \end{aligned} \end{aligned}$$

(3.12)

for any finite $\widetilde{q}_2 >\widetilde{q}_1$ and $\widetilde{r}_2> \widetilde{r}_1$ with $\frac{1}{\widetilde{q}_1} > \frac{1}{q_2}+\frac{1}{\widetilde{q}_2}$.

It follows from (3.10) and (3.11) that by taking $\alpha \rightarrow 0$, we can take $(q_1, r_1)$ (and $(\widetilde{q}_1, \widetilde{r}_1)$, respectively) arbitrarily close to (q, r) (and to $(\widetilde{q}, \widetilde{r})$, respectively). Moreover, by taking $\alpha > 0$ sufficiently small, we can also take $(\widetilde{q}_2,\widetilde{r}_2)$ arbitrarily close to $(\widetilde{q}_1,\widetilde{r}_1)$. Hence, from (3.1), we can guarantee

$$\begin{aligned} 2\widetilde{q}_2< q_1 \quad \text {and} \quad 2\widetilde{r}_2< r_1 \end{aligned}$$

(3.13)

by taking $\alpha > 0$ sufficiently small. Therefore, from (3.8) and (3.12) with (3.9) and (3.13), we obtain

$$\begin{aligned} \big \Vert \Phi ^N(v_N) \big \Vert _ {X^s(T)} \lesssim \Vert (u_0, u_1)\Vert _{\mathcal {H}^s} + T^\theta \big \Vert \Theta _N \big \Vert _{L^{q_2}_T W^{-\alpha ,\infty }_x}\Big (1+\beta \Vert v_N \Vert _{X^s(T)}\Big ). \end{aligned}$$

(3.14)

Proceeding as in (3.8) and (3.12), we have

$$\begin{aligned} \big \Vert \Phi ^N(v_N)-\Phi ^N(w_N) \big \Vert _ {X^s(T)} \lesssim T^{\theta }\big \Vert \Theta _N \big \Vert _{L^{q_2}_{T} W^{-\alpha ,\infty }_x} \Vert e^{i\beta v_N} -e^{i\beta w_N} \Vert _{L^{\widetilde{q}_2}_T W^{\alpha ,\widetilde{r}_2}_x}. \end{aligned}$$

(3.15)

As for the last factor in (3.15), by applying the mean value theorem with $F(u)= e^{i\beta u}$, the fractional Leibniz rule (3.4) in Lemma 3.1, and the fractional chain rule (Lemma 3.2), we have

$$\begin{aligned} \begin{aligned}&\Vert e^{i\beta v_N} -e^{i\beta w_N} \Vert _{L^{\widetilde{q}_2}_T W^{\alpha ,\widetilde{r}_2}_x}\\&\quad = \bigg \Vert (v_N-w_N)\int _0^1F'(\tau v_N + (1-\tau )w_N)d\tau \bigg \Vert _{L^{\widetilde{q}_2}_T W^{\alpha ,\widetilde{r}_2}_x} \\&\quad \lesssim \Vert v_N-w_N\Vert _{L^{2\widetilde{q}_2}_T W^{\alpha ,2\widetilde{r}_2}_x}\int _0^1\Vert F'(\tau v_N + (1-\tau )w_N)\Vert _{L^{2\widetilde{q}_2}_T W^{\alpha ,2\widetilde{r}_2}_x}d\tau \\&\quad \lesssim \beta \Vert v_N-w_N\Vert _{L^{2\widetilde{q}_2}_T W^{\alpha ,2\widetilde{r}_2}_x} \Big (1+\beta \Vert v_N\Vert _{L^{2\widetilde{q}_2}_T W^{\alpha ,2\widetilde{r}_2}_x}+\beta \Vert w_N\Vert _{L^{2\widetilde{q}_2}_T W^{\alpha ,2\widetilde{r}_2}_x}\Big ). \end{aligned} \end{aligned}$$

(3.16)

Hence, from (3.15) and (3.16), we obtain

$$\begin{aligned} \begin{aligned} \big \Vert \Phi ^N(v_N)-\Phi ^N(w_N) \big \Vert _ {X^s(T)}&\lesssim T^{\theta }\beta (1+\beta R)\big \Vert \Theta _N \big \Vert _{L^{ q_2}_{T} W^{-\alpha ,\infty }_x} \Vert v_N-w_N\Vert _{X^s(T)} \end{aligned} \end{aligned}$$

(3.17)

for $v_N, w_N \in B_R \subset X^s(T)$.

Let $\alpha > 0$ be sufficiently small as above. Then, Proposition 1.1 states that, given any finite $p\ge 1$, $\{\Theta _N\}_{N \in \mathbb {N}}$ is uniformly bounded in $L^p(\Omega ;L^{q_2}([0,T_0];W^{-\alpha ,\infty }(\mathbb {T}^2)))$, provided that

$$\begin{aligned} 0< T_0 < \frac{8\pi \alpha }{\beta ^2}. \end{aligned}$$

(3.18)

Fix $T_0 > 0$ satisfying (3.18). Given $N \in \mathbb {N}$ and $\lambda > 0$, define $\Omega _{N, \lambda }$ by

$$\begin{aligned} \Omega _{N, \lambda } = \Big \{\omega \in \Omega : \big \Vert \Theta _N \big \Vert _{L^{ q_2}_{T_0} W^{-\alpha ,\infty }_x}< \lambda \Big \}. \end{aligned}$$

On $\Omega _{N, \lambda }$, it follows from (3.14) and (3.17) that $\Phi ^N$ is a contraction on the ball $B_R\subset X^s(T)$, where $R\sim \Vert (u_0, u_1)\Vert _{\mathcal {H}^s}$ and $T = T(R, \lambda , \beta ) = T\big (\Vert (u_0, u_1)\Vert _{\mathcal {H}^s}, \lambda , \beta \big ) > 0$ such that $T \le T_0$. Note that we can choose $T>0$ such that

$$\begin{aligned} T \sim _{R, \beta } (1+\lambda )^{-\kappa } \end{aligned}$$

(3.19)

for some $\kappa > 0$ and any $\lambda \gg 1$.

By Proposition 1.1 and Chebyshev’s inequality, we have

$$\begin{aligned} \sup _{N \in \mathbb {N}} P\big ( \Omega _{N, \lambda }^c\big ) = o(1) \end{aligned}$$

(3.20)

as $\lambda \rightarrow \infty$. This proves local well-posedness of (3.7) uniformly in $N \in \mathbb {N}$. Here, the uniformity refers to the fact that, given any small $T>0$, we have a uniform (in N) control (i.e. a uniform lower bound in terms of T) on the probabilities of the sets $\Omega _N(T) = \Omega _{N, \lambda (T)}$, where well-posedness of (3.7) holds on [0, T], thanks to (3.19) and (3.20).

Let $\Theta$ be the limit of $\Theta _N$ in $L^p(\Omega ;L^{q_2}([0,T_0];W^{-\alpha ,\infty }(\mathbb {T}^2)))$ constructed in Proposition 1.1. Define $\Omega _{\lambda }$ by

$$\begin{aligned} \Omega _{ \lambda } = \Big \{\omega \in \Omega : \Vert \Theta \Vert _{L^{ q_2}_{T_0} W^{-\alpha ,\infty }_x}< \lambda \Big \}. \end{aligned}$$

(3.21)

Then, by repeating the argument above, we see that the limiting equation (1.20) for v, written in the Duhamel formulation as

$$\begin{aligned} v(t) = \partial _tS(t)u_0 + S(t)u_1 -\int _0^tS(t-t'){\hbox {Im}}\big (\Theta e^{i\beta v}\big )(t')dt', \end{aligned}$$

(3.22)

is well-posed on the time interval [0, T], where $T = T(R, \lambda , \beta ) < T_0$ satisfies (3.19). In view of Proposition 1.1, we then conclude that there exists an almost surely positive stopping time

$$\begin{aligned} \tau = \tau \big (\Vert (u_0, u_1)\Vert _{\mathcal {H}^s}, \beta \big ) \sim C(\Vert (u_0, u_1)\Vert _{\mathcal {H}^s}, \beta ) \big (1 + \Vert \Theta \Vert _{L^{ q_2}_{T_0} W^{-\alpha ,\infty }_x}\big )^{-\kappa } \end{aligned}$$

(3.23)

such that the limiting equation (1.20) for v is well-posed on the time interval $[0, \tau ]$.

In the following, fix sufficiently small $T = T\big (\Vert (u_0, u_1)\Vert _{\mathcal {H}^s}, \beta \big ) > 0$, satisfying (3.18) (with $T_0$ replaced by T) and set $\Omega (T) = \{ \tau \ge T\}$, where $\tau$ is defined in (3.23). Namely, for any $\omega \in \Omega (T)$, the limiting equation (3.22) is well-posed on [0, T] by the argument above. Moreover, in view of (3.21) and (3.23), we may assume that $\Omega (T) = \Omega _\lambda$ for some $\lambda = \lambda (T) >0$ satisfying (3.19). Note that $P(\Omega (T)) > 0$ for any sufficiently small $T>0$ and $P(\Omega (T)) \rightarrow 1$ as $T \rightarrow 0$. In the following, we work with the conditional probability $P_T$ given the event $\Omega (T)$, defined by

$$\begin{aligned} P_T (A) = P(A|\Omega (T)) = \frac{P(A \cap \Omega (T))}{P(\Omega (T))}. \end{aligned}$$

We first check that the truncated dynamics (3.7) is well-posed on [0, T] outside a set of $P_T$-probability o(1) as $N \rightarrow \infty$. Let $N \in \mathbb {N}$. By defining a set $\Sigma _N$ by

$$\begin{aligned} \Sigma _{N} = \Big \{\omega \in \Omega : \big \Vert \Theta _N - \Theta \big \Vert _{L^{ q_2}_{T_0} W^{-\alpha ,\infty }_x}\le 1 \Big \}, \end{aligned}$$

we have

$$\begin{aligned} \Omega (T) \cap \Sigma _N = \Omega _\lambda \cap \Sigma _N \subset \Omega _{N, \lambda +1} . \end{aligned}$$

Without loss of generality, we may assume that the Eq. (3.7) is also well-posed on [0, T] for any $\omega \in \Omega (T) \cap \Sigma _N$. Furthermore, since $\Theta _N$ converges in probability to $\Theta$ in $L^{q_2}([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2))$, we have $P(\Sigma _N^c) \rightarrow 0$ as $N \rightarrow \infty$. Namely, we have

$$\begin{aligned} P_T(\Omega (T) \cap \Sigma _N) \longrightarrow 1, \end{aligned}$$

(3.24)

as $N \rightarrow \infty$. This verifies well-posedness of the truncated dynamics (3.7) on the time interval [0, T] asymptotically $P_T$-almost surely in the sense of (3.24).

Given $(u_0, u_1) \in \mathcal {H}^s(\mathbb {T}^2)$ and $\Theta _0 \in L^{q_2}([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2)))$, define a map $\Phi = \Phi _{(u_0, u_1), \Theta _0}$ by

$$\begin{aligned} \Phi (v)(t) = \partial _tS(t)u_0 + S(t)u_1 -\int _0^tS(t-t'){\hbox {Im}}\big ( \Theta _0 e^{i\beta v}\big )(t')dt'. \end{aligned}$$

(3.25)

Then, a slight modification of the analysis presented above shows that the map:

$$\begin{aligned} \big ((u_0, u_1) , \Theta _0\big ) \in \mathcal {H}^s(\mathbb {T}^2)\times L^{q_2}([0,T];W^{-\alpha ,\infty }(\mathbb {T}^2))) \longrightarrow v \in X^s(T) \end{aligned}$$

is continuous, where $v = \Phi (v)$ is the unique fixed point for $\Phi$ and $0 < T \le T_0$ is sufficiently small. From this observation and convergence in probability of $\Theta _N$ to $\Theta$ deduced from Proposition 1.1, we conclude that the solution $v_N$ to (3.25) converges in probability with respect to the conditional probability $P_T$ to v in $X^s(T)$.¹³

Finally, recalling the decompositions

$$\begin{aligned} u_N=\Psi _N + v_N \quad \text {and} \quad u = \Psi + v \end{aligned}$$

and from Lemma 2.6 that $\Psi _N$ converges P-almost surely (and hence $P_T$-almost surely) to $\Psi$ in $C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2))$, we conclude that $u_N$ converges to u, in $C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2))$, in probability with respect to the conditional probability $P_T$. This completes the proof of Theorem 1.2.

3.3 Triviality of the unrenormalized model

We conclude this paper by establishing a triviality result for the unrenormalized model (Proposition 1.4), whose proof follows from a small modification of the proof of Theorem 1.2.

The main idea is to follow the idea from [23, 33, 35] and artificially introduce a renormalization constant $\gamma _N(t)$ in (1.18). Note that unlike the previous work [23, 33, 35], our renormalization constant appears in a multiplicative manner, which makes our analysis particularly simple. Start with the truncated equation (1.21) and rewrite it as

$$\begin{aligned} \partial _t^2u_N + (1-\Delta ) u_N +\gamma _N^{-1} {\hbox {Im}}\big (\gamma _N e^{i\beta u_N}\big ) = \mathbf {P}_N\xi . \end{aligned}$$

With the decomposition

$$\begin{aligned} u_N = \Psi _N + v_N, \end{aligned}$$

(3.26)

it follows from (1.16) and (1.18) that $\gamma _N e^{i\beta u_N} = \Theta _N e^{i \beta v_N}$. Then, we may repeat the analysis presented in the previous subsection and establish local well-posedness for the equation satisfied by $v_N$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2v_N + (1-\Delta )v_N + \gamma _N^{-1} {\hbox {Im}}\big (\Theta _N e^{i\beta v_N}\big )=0\\ (v_N,\partial _tv_N)\big |_{t=0}=(u_0,u_1). \end{array}\right. } \end{aligned}$$

(3.27)

Noting that $0 \le \gamma _N^{-1}(t) \le 1$, we once again have a uniform (in N) control on the probabilities of the sets $\Omega _N(T) = \Omega _{N, \lambda (T)}$, where well-posedness of (3.27) holds on [0, T]. Moreover, we have

$$\begin{aligned} \Vert v_N \Vert _{X^s(T)} \lesssim \Vert (u_0, u_1)\Vert _{\mathcal {H}^s} \end{aligned}$$

(3.28)

on $\Omega _N(T)$.

By writing (3.27) in the Duhamel formulation:

$$\begin{aligned} v_N(t)= \partial _tS(t)u_0 + S(t)u_1 -\int _0^tS(t-t')\gamma _N^{-1}(t'){\hbox {Im}}\big (\Theta _Ne^{i\beta v_N}\big )(t')dt', \end{aligned}$$

we see from (3.8) and (3.12) with (3.28) that

$$\begin{aligned} \begin{aligned} \Vert v_N-\partial _tS(t)u_0 - S(t)u_1\Vert _{X^s(T)}&\lesssim T^\theta \big \Vert \gamma _N^{-1}(t) \Theta _N \big \Vert _{L^{q_2}_T W^{-\alpha ,\infty }_x}\Big (1+\beta \Vert v_N \Vert _{X^s(T)}\Big )\\&\lesssim \Vert \gamma _N^{-1}(t) \Vert _{L^{\tfrac{1}{\delta }}_T}\big \Vert \Theta _N \big \Vert _{L^{\frac{q_2}{1-\delta q_2}}_{T_0} W^{-\alpha ,\infty }_x}\\&\quad \times \Big (1+\beta \Vert (u_0,u_1)\Vert _{\mathcal {H}^s}\Big ) \end{aligned} \end{aligned}$$

(3.29)

for any small $\delta >0$. From (1.18) with (1.12) and Lemma 2.7, we have

$$\begin{aligned} \Vert \gamma _N^{-1}(t) \Vert _{L^{\tfrac{1}{\delta }}_T} \lesssim \Vert e^{-\frac{\beta ^2t}{8\pi }\log N}\Vert _{L^{\tfrac{1}{\delta }}_T}\le C(\beta ,\delta )\big (\log N\big )^{-1} \longrightarrow 0, \end{aligned}$$

(3.30)

as $N \rightarrow \infty$. Then, it follows from (3.29), (3.30), Proposition 1.1, and Chebyshev’s inequality that $v_N -\partial _tS(t) u_0 - S(t)u_1$ converges in probability to 0 in $X^s(T)$ as $N \rightarrow \infty$. Recalling the decomposition (3.26) and the almost sure convergence of $\Psi _N$ to $\Psi$ in $C([0,T];W^{-\varepsilon ,\infty }(\mathbb {T}^2))$ (Lemma 2.6), we conclude that $u_N$ converges in probability to $\partial _tS(t)u_0 + S(t) u_1 + \Psi$, which is the unique solution to the linear stochastic wave equation (1.22) with initial data $(u_0, u_1)$. This completes the proof of Proposition 1.4.