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1 Introduction↩︎

1.1 Hyperbolic \(O(N)\) linear sigma model and the Smoluchowski-Kramers approximation↩︎

In this paper, we study the following coupled system of stochastic damped nonlinear wave equations (SdNLW) on the two-dimensional torus \(\mathbb{T}^2 = (\mathbb{R}/ 2 \pi \mathbb{Z})^2\) with a parameter \(0 < \varepsilon\leq 1\): \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^{N, j} = - \frac{1}{N} \sum_{k = 1}^N (u_\varepsilon^{N, k})^2 u_\varepsilon^{N, j} + \sqrt{2} \xi^j , \qquad j = 1 , \dots, N , \label{NLWNe0} \end{align}\tag{1}\]

where \(\xi^j\)’s are independent real-valued space-time white noises on \(\mathbb{R}_+ \times \mathbb{T}^2\). When \(\varepsilon= 1\), the system 1 is also referred to as the hyperbolic \(O(N)\) linear sigma model, whose study was initiated by Oh, the second author, and the third author in [1]. For later convenience, we address the system 1 by the abbreviation \(\text{HLSM}_{\varepsilon, N}\), which stands for the hyperbolic \(O(N)\) linear sigma model indexed by a parameter \(\varepsilon\). Our goal is to show that each component \(u_\varepsilon^{N, j}\) of the solution to 1 converges globally in time to the solution \(u^j\) of the stochastic nonlinear heat equation (SNLH) with a mean-field nonlinearity: \[\begin{align} (\partial_t+ 1 - \Delta) u^j = - \mathbb{E}[(u^j)^2] u^j + \sqrt{2} \xi^j . \label{NLH0} \end{align}\tag{2}\]

We exploit such convergence via the following two regimes:

1. We first let \(\varepsilon\to 0\) to obtain \(u^{N, j}\) with \(1 \leq j \leq N\) satisfying the following coupled system of SNLH, which we also refer to as the parabolic \(O(N)\) linear sigma model (\(\text{PLSM}_N\)): \[\begin{align} (\partial_t+ 1 - \Delta) u^{N, j} = - \frac{1}{N} \sum_{k = 1}^N (u^{N, k})^2 u^{N, j} + \sqrt{2} \xi^j . \label{NLHN0} \end{align}\tag{3}\]

   We then take \(N \to \infty\) to obtain \(u^j\) satisfying 2 .

2. We first let \(N \to \infty\) to obtain \(u_\varepsilon^j\) satisfying the following mean-field SdNLW indexed by a parameter \(0 < \varepsilon\leq 1\) (mean-field \(\text{SdNLW}_\varepsilon\)): \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^j = - \mathbb{E}[(u_\varepsilon^j)^2] u_\varepsilon^j + \sqrt{2} \xi^j . \label{NLWe0} \end{align}\tag{4}\]

   We then take \(\varepsilon\to 0\) to obtain \(u^j\) satisfying 2 .

As a result, we obtain the following commutative diagram for global-in-time dynamics:

The process of taking \(\varepsilon\to 0\) is called the Smoluchowski-Kramers approximation [2], [3], which is the main topic in this paper. The process of taking \(N \to \infty\) is called the mean-field limit, which has been studied in [4] for SNLH and [1] for SdNLW.

Let us first discuss some background on the hyperbolic \(O(N)\) linear sigma model and the Smoluchowski-Kramers approximation. In Euclidean quantum field theory, models are described by probability measures on spaces of distributions (see [5] for a more detailed introduction). The stochastic quantization paradigm, introduced by Parisi-Wu in [6], studies such measures by constructing a stochastic partial differential equation (SPDE) whose unique invariant measure is the target quantum field theory measure. One of the most extensively studied examples is the scalar-valued \(\Phi^4_2\)-model (see 129 in Appendix 6 with \(N = 1\)), where the superscript “4” denotes the quartic interaction and the subscript “2” the spatial dimension. Its parabolic stochastic quantization is given by the SNLH 3 with \(N = 1\), whose local well-posedness theory was established by Da Prato-Debussche in [7] and global theory by Mourrat-Weber in [8]; see also [9] for a unique ergodicity result. While the scalar-valued case is fundamental, many physically relevant models involve fields taking values in an \(N\)-dimensional target space. A particular example is the \(N\)-component \(\Phi^4_2\)-model introduced by Wilson in [10], given by 129 , which is also known as the \(O(N)\) linear sigma model. The term “linear” refers to the fact that the target space \(\mathbb{R}^N\) is a linear space. The corresponding parabolic stochastic quantization is given by the \(\mathrm{PLSM}_N\) 3 , which was studied by Shen-Smith-Zhu-Zhu in [4]. See also [11] for the (spatial) 3-dimensional setting and also a nice survey paper [12].

In addition to the parabolic approach, there has been significant progress in the study of hyperbolic stochastic quantization, also known as canonical stochastic quantization or Hamiltonian stochastic quantization; see [13]. For the scalar \(\Phi^4_2\)-model, the corresponding hyperbolic stochastic quantization is described by the SdNLW 1 with \(N = 1\) and \(\varepsilon= 1\). Local well-posedness for this equation was established by Gubinelli-Koch-Oh in [14], while global well-posedness was subsequently proved by Gubinelli-Koch-Oh-Tolomeo in [15]; see also [16] for a unique ergodicity result. More recently, the hyperbolic framework has been extended beyond the scalar setting. In [1], Oh, the second author, and the third author initiated the study of the \(N\)-component model by introducing the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 with \(\varepsilon= 1\), providing a hyperbolic stochastic quantization of the \(O(N)\) linear sigma model.

In the present work, we investigate the convergence of the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 to the \(\mathrm{PLSM}_N\) 3 in the limit of a vanishing second-order (mass) term (\(\varepsilon\to 0\)), alongside their mean-field convergence. This zero-mass limit is known as the Smoluchowski-Kramers approximation and has been studied across various finite and infinite dimensional settings; see [17]–[35]. In the context of stochastic quantization of the \(\Phi^4_2\)-model, such approximation has been studied by Fukuizumi-Hoshino-Inui in [36] and by Zine in [37]. From an analytical perspective, the approximation connects two fundamentally different analytical regimes: a parabolic dynamics driven by the strong instantaneous smoothing of the heat semigroup and a hyperbolic dynamics governed by dispersive and Hamiltonian effects. From a computational perspective, it links overdamped and underdamped Langevin dynamics, balancing the improved state-space exploration of the latter against the greater numerical stability of the former; see [38]–[40]. In particular, it provides a mathematical justification for treating \(\varepsilon\) as a tuning parameter without changing the underlying target quantum field theory measure.

The present work naturally extends [37], where the author treated the scalar case \(N = 1\). The presence of the additional parameter \(N\) leads to a natural question: do the limits \(\varepsilon\to 0\) and \(N \to \infty\) commute? At first glance, the two limiting procedures appear largely independent, since \(\varepsilon\) enters only through the second-order time derivative, whereas \(N\) appears only in the nonlinearity. Indeed, we are able to prove the convergence in either order to the unique solution of the limiting equation (the mean-field SNLH 2 ); see Figure 1. Nevertheless, one of the main findings of this paper is that, although both convergence routes ultimately lead to the same limiting equation, the analysis of the limit \(\varepsilon\to 0\) for global-in-time dynamics differs substantially depending on the order in which the limits are taken. We provide a more detailed discussion in Subsection 1.2. Roughly speaking, this distinction stems from the fact that the coupled systems 1 and 3 are defined pathwise, for each \(\omega\) in a full-measure subset of \(\Omega\), whereas the mean-field equations 2 and 4 involve averaging over the entire probability space.

1.2 Main results on convergence of dynamics↩︎

Before stating the main results regarding the commutative diagram in Figure 1, let us take a closer look at the four equations 1 , 2 , 3 , and 4 .

Let us first look at \(\mathrm{HLSM}_{\varepsilon, N}\) 1 . Given \(j \in \mathbb{N}\) and \(0 < \varepsilon\leq 1\), we define \(\Psi_\varepsilon^j\) as the stochastic convolution satisfying the following linear stochastic damped wave equation with a parameter \(\varepsilon\) and zero initial data: \[\begin{align} \begin{cases} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) \Psi_\varepsilon^j = \sqrt{2} \xi^j \\ (\Psi_\varepsilon^j, \partial_t\Psi_\varepsilon^j) |_{t = 0} = (0, 0) . \end{cases} \label{Psije} \end{align}\tag{5}\]

As can be seen from 37 below, the frequency truncated version of \(\Psi_\varepsilon^j\) has variance that diverges as the truncation is removed. This means that the stochastic convolution \(\Psi_\varepsilon^j\) is almost surely a distribution but in general not a function on \(\mathbb{T}^2\), and so we need to make sense of its powers through renormalization.

Let us proceed as in [1], [14], [15], [37] with the following first order expansion (see [41], [42]), also known as the Da Prato-Debussche trick in the parabolic setting (see [7]): \[\begin{align} u_\varepsilon^{N, j} = \Psi_\varepsilon^j + v_\varepsilon^{N, j} , \qquad j = 1, \dots, N , \label{expNe} \end{align}\tag{6}\]

where \(v_\varepsilon^{N, j}\) is a remainder term satisfying \[\begin{align} \begin{aligned} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) v_\varepsilon^{N, j} &= - \frac{1}{N} \sum_{k = 1}^N \big( :\!{ (\Psi_\varepsilon^k)^2 \Psi_\varepsilon^j }\!: + :\!{ (\Psi_\varepsilon^k)^2 }\!: v_\varepsilon^{N, j} + 2 v_\varepsilon^{N, k} :\!{ \Psi_\varepsilon^k \Psi_\varepsilon^j }\!: \\ &\quad + 2 \Psi_\varepsilon^k v_\varepsilon^{N, k} v_\varepsilon^{N, j} + (v_\varepsilon^{N, k})^2 \Psi_\varepsilon^j + (v_\varepsilon^{N, k})^2 v_\varepsilon^{N, j} \big) , \qquad j = 1, \dots, N. \end{aligned} \label{NLWNev} \end{align}\tag{7}\]

On the right-hand side of 7 , we have already applied Wick renormalization to the products and powers of \(\Psi_\varepsilon^j\)’s. See 38 and 39 for the precise definitions, but at the moment we make the following interpretation: \[\begin{align} :\!{ \Psi_\varepsilon^k \Psi_\varepsilon^j }\!: &= \begin{cases} :\!{(\Psi_\varepsilon^j)^2}\!: & \text{if } k = j \\ \Psi_\varepsilon^k \Psi_\varepsilon^j & \text{if } k \neq j , \end{cases} \\ :\!{ (\Psi_\varepsilon^k)^2 \Psi_\varepsilon^j }\!: &= \begin{cases} :\!{(\Psi_\varepsilon^j)^3}\!: & \text{if } k = j \\ :\!{(\Psi_\varepsilon^k)^2}\!: \Psi_\varepsilon^j & \text{if } k \neq j , \end{cases} \end{align}\]

where \(:\!{(\Psi_\varepsilon^j)^2}\!:\) and \(:\!{(\Psi_\varepsilon^j)^3}\!:\) denote the standard Wick powers. Then, by setting \[\begin{align} :\!{ (u_\varepsilon^{N, k})^2 u_\varepsilon^{N, j} }\!: \, \stackrel{\mathrm{def}}{=}&\big( :\!{ (\Psi_\varepsilon^k)^2 \Psi_\varepsilon^j }\!: + :\!{ (\Psi_\varepsilon^k)^2 }\!: v_\varepsilon^{N, j} + 2 v_\varepsilon^{N, k} :\!{ \Psi_\varepsilon^k \Psi_\varepsilon^j }\!: \\ &\quad + 2 \Psi_\varepsilon^k v_\varepsilon^{N, k} v_\varepsilon^{N, j} + (v_\varepsilon^{N, k})^2 \Psi_\varepsilon^j + (v_\varepsilon^{N, k})^2 v_\varepsilon^{N, j} \big) , \end{align}\]

we get the following renormalized version of \(\mathrm{HLSM}_{\varepsilon, N}\) 1 : \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^{N, j} = - \frac{1}{N} \sum_{k = 1}^N :\!{ (u_\varepsilon^{N, k})^2 u_\varepsilon^{N, j} }\!: + \, \sqrt{2} \xi^j , \qquad j = 1, \dots, N. \label{NLWNeu} \end{align}\tag{8}\]

As in [1], [14], [15], we say that \(\mathbf{u}_\varepsilon^N = (u_\varepsilon^{N, j})_{1 \leq j \leq N}\) is a solution to the renormalized \(\mathrm{HLSM}_{\varepsilon, N}\) 8 if \(u_\varepsilon^{N, j}\) is given by 6 with \(\mathbf{v}_\varepsilon^N = (v_\varepsilon^{N, j})_{1 \leq j \leq N}\) being a solution to the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 7 .

In order to discuss the convergence problem later on, we need to know that the equation 8 does have a solution. This is provided by the following proposition. For notations, we write \[\begin{align} \mathcal{H}^s (\mathbb{T}^2) = H^s (\mathbb{T}^2) \times H^{s - 1} (\mathbb{T}^2) , \end{align}\]

where \(H^s\) is the \(L^2\)-based Sobolev space with regularity \(s \in \mathbb{R}\). Given \(N \in \mathbb{N}\) and a function space \(X\), we write \(X^{\otimes N}\) as the \(N\)-fold product of \(X\) with itself: \(X \times \cdots \times X\).

The proof of Proposition 1 is similar to that of [1], which we now briefly describe. Our goal here is to prove global well-posedness of the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 7 for \((\mathbf{v}_\varepsilon^N, \partial_t\mathbf{v}_\varepsilon^N) = ((v_\varepsilon^{N, j}, \partial_tv_\varepsilon^{N, j}))_{1 \leq j \leq N}\) with initial data \((\mathbf{u}_0^{N}, \mathbf{u}_1^{N})\). In view of the one degree of smoothing from the linear damped wave dynamics from 28 (we allow dependence on \(\varepsilon\) here) and the regularity properties of the Wick products and powers of the stochastic convolution \(\Psi_\varepsilon^j\) in Lemma 8, we can prove pathwise local well-posedness for 7 using the same framework as in [1]. Note that the proof does not use any auxiliary function space and so the uniqueness of a solution \((\mathbf{v}_\varepsilon^N, \partial_t\mathbf{v}_\varepsilon^N)\) holds in the entire space \(C ([0, T] ; \mathcal{H}^s (\mathbb{T}^2))^{\otimes N}\) for some almost surely positive time \(T\).

To achieve global well-posedness, we aim to use an energy functional to establish a global-in-time a priori bound for the solution, which allows us to iterate the local well-posedness argument. Given a tuple of space-time functions \((\mathbf{w}^N, \partial_t\mathbf{w}^N) = ((w^{N, j}, \partial_tw^{N, j}))_{1 \leq j \leq N}\), we define the energy functional \(E_\varepsilon^N (\mathbf{w}^N, \partial_t\mathbf{w}^N)\) by \[\begin{align} \begin{aligned} E_\varepsilon^N (\mathbf{w}^N, \partial_t\mathbf{w}^N) &\stackrel{\mathrm{def}}{=}\frac{1}{2} \sum_{j = 1}^N \int_{\mathbb{T}^2} \big( (\varepsilon\partial_tw^{N, j})^2 + (w^{N, j})^2 + |\nabla w^{N, j}|^2 \big) dx \\ &\quad + \frac{1}{4N} \int_{\mathbb{T}^2} \Big( \sum_{j = 1}^N (w^{N, j})^2 \Big)^2 dx . \end{aligned} \label{defEeN} \end{align}\tag{9}\]

It is not hard to check that \(E_\varepsilon^N (\mathbf{w}^N, \partial_t\mathbf{w}^N)\) is conserved under the flow of the following coupled NLW system: \[\begin{align} (\varepsilon^2 \partial_t^2 + 1 - \Delta) w^{N, j} = - \frac{1}{N} \sum_{k = 1}^N (w^{N, k})^2 w^{N, j} , \qquad j = 1, \dots, N . \label{NLWsys} \end{align}\tag{10}\]

In fact, we can write 10 as the following Hamiltonian formulation: \[\begin{align} \varepsilon^2 \partial_t\begin{pmatrix} \mathbf{w}^N \\ \partial_t\mathbf{w}^N \end{pmatrix} = \begin{pmatrix} 0 & \boldsymbol{Id}^N \\ - \boldsymbol{Id}^N & 0 \end{pmatrix} \nabla_{(L^2 (\mathbb{T}^2))^{\otimes 2N}} E_\varepsilon^N (\mathbf{w}^N, \partial_t\mathbf{w}^N) , \label{Hamil} \end{align}\tag{11}\]

where \(\boldsymbol{Id}^N\) denotes the \(N \times N\) identity matrix and \(\nabla_{(L^2 (\mathbb{T}^2))^{\otimes 2N}}\) denotes the Fréchet derivative with respect to the norm \((L^2 (\mathbb{T}^2))^{\otimes 2N} = L^2 (\mathbb{T}^2) \times \cdots \times L^2 (\mathbb{T}^2)\). As in the situation in [15], there are two difficulties in proving pathwise global well-posedness for 7 . Firstly, due to the roughness of the stochastic objects in 7 , the local-in-time solution \((\mathbf{v}_\varepsilon^N (t), \partial_t\mathbf{v}_\varepsilon^N (t))\) does not belong to the energy space \((\mathcal{H}^1 (\mathbb{T}^2))^{\otimes N}\). Secondly, the energy \(E_\varepsilon^N (\mathbf{v}_\varepsilon^N, \partial_t\mathbf{v}_\varepsilon^N)\) is not conserved under the flow of the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 7 , even if we assume that \((\mathbf{v}_\varepsilon^N (t), \partial_t\mathbf{v}_\varepsilon^N (t)) \in (\mathcal{H}^1 (\mathbb{T}^2))^{\otimes N}\). The above issues have been solved by Gubinelli-Koch-Oh-Tolomeo in [15] via a hybrid approach combining the \(I\)-method [43] (for dealing with the first issue) and a Gronwall-type argument [44] (for dealing with the second issue). This hybrid approach was adapted to the vector-valued setting in [1]. With minor modifications to allow for dependence on \(\varepsilon\), this argument also applies to proving pathwise global well-posedness for 7 .

For the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 7 , we now take \(\varepsilon\to 0\) and \(N \to \infty\) to formally obtain the limiting equation: \[\begin{align} (\partial_t+ 1 - \Delta) v^j = - 2 \mathbb{E}[\Psi_0^j v^j] \Psi_0^j - 2 \mathbb{E}[\Psi_0^j v^j] v^j - \mathbb{E}[(v^j)^2] \Psi_0^j - \mathbb{E}[(v^j)^2] v^j , \label{NLHv} \end{align}\tag{12}\]

where \(\Psi_0^j\) is the stochastic convolution satisfying the following linear stochastic heat equation with zero initial data: \[\begin{align} \begin{cases} (\partial_t+ 1 - \Delta) \Psi_0^j = \sqrt{2} \xi^j \\ \Psi_0^j |_{t = 0} = 0 . \end{cases} \label{Psij0} \end{align}\tag{13}\]

By writing \[\begin{align} u^j = \Psi_0^j + v^j , \label{exp} \end{align}\tag{14}\]

we can write out the following renormalized version of the mean-field SNLH 2 : \[\begin{align} (\partial_t+ 1 - \Delta) u^j = - \mathbb{E}[(u^j)^2 - (\Psi_0^j)^2] u^j + \sqrt{2} \xi^j. \label{NLHu} \end{align}\tag{15}\]

As before, we say that \(u^j\) is a solution to the renormalized mean-field SNLH 15 if \(u^j\) is given by 14 with \(v^j\) being a solution to the perturbed mean-field SNLH 12 .

We also have the following well-posedness result for the equation 15 .

Global well-posedness for the renormalized mean-field SNLH was established in [4], but in a different formulation from that of our Proposition 2. The initial condition in [4] is \(u_0^j \in L^4 (\Omega; L^4 (\mathbb{T}^2))\) and a unique global-in-time solution to the perturbed mean-field SNLH 12 exists in the space \[\begin{align} L^2 (\Omega; C ((0, T]; \mathcal{C}^\beta(\mathbb{T}^2)) \cap C([0, T]; L^4 (\mathbb{T}^2)) ) \end{align}\]

for any \(T > 0\) and some sufficiently small \(\beta> 0\), where \(\mathcal{C}^s (\mathbb{T}^2)\) denotes the Besov-Hölder space on \(\mathbb{T}^2\) with regularity \(s\) (see Subsection 2.1). In this paper, we choose to treat the wave equations and the heat equations in a uniform manner by putting the initial data and the solutions in \(L^2\)-based Sobolev spaces with more or less the same regularity. In order to obtain the desired regularity as claimed in Proposition 2, we need some additional treatment based on the result in [4], which we present in Subsection 3.2.

We now take a closer look at the two regimes of the convergence problem, starting with regime I. For the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 7 , we fix \(N \in \mathbb{N}\) and take \(\varepsilon\to 0\) to formally obtain the following perturbed \(\text{PLSM}_N\): \[\begin{align} \begin{aligned} (\partial_t+ 1 - \Delta) v^{N, j} = - \frac{1}{N} &\sum_{k = 1}^N \big( :\!{ (\Psi_0^k)^2 \Psi_0^j }\!: + :\!{ (\Psi_0^k)^2 }\!: v^{N, j} + 2 v^{N, k} :\!{ \Psi_0^k \Psi_0^j }\!: \\ &\, + 2 \Psi_0^k v^{N, k} v^{N, j} + (v^{N, k})^2 \Psi_0^j + (v^{N, k})^2 v^{N, j} \big) , \qquad j = 1, \dots, N. \end{aligned} \label{NLHNv} \end{align}\tag{16}\]

By writing \[\begin{align} u^{N, j} = \Psi_0^j + v^{N, j} , \qquad j = 1, \dots, N , \label{expN} \end{align}\tag{17}\]

and setting \[\begin{align} \sum_{k = 1}^N :\!{ (u^{N, k})^2 u^{N, j} }\!: \, \stackrel{\mathrm{def}}{=}&\sum_{k = 1}^N \big( :\!{ (\Psi_0^k)^2 \Psi_0^j }\!: + :\!{ (\Psi_0^k)^2 }\!: v^{N, j} + 2 v^{N, k} :\!{ \Psi_0^k \Psi_0^j }\!: \\ &\quad + 2 \Psi_0^k v^{N, k} v^{N, j} + (v^{N, k})^2 \Psi_0^j + (v^{N, k})^2 v^{N, j} \big) , \end{align}\]

we get the following renormalized version of the \(\text{PLSM}_N\): \[\begin{align} (\partial_t+ 1 - \Delta) u^{N, j} = - \frac{1}{N} \sum_{k = 1}^N :\!{ (u^{N, k})^2 u^{N, j} }\!: + \, \sqrt{2} \xi^j , \qquad j = 1, \dots, N. \label{NLHNu} \end{align}\tag{18}\]

As before, we say that \(\mathbf{u}^N = (u^{N, j})_{1 \leq j \leq N}\) is a solution to the renormalized \(\mathrm{PLSM}_N\) 18 if \(u^{N, j}\) is given by 17 with \(\mathbf{v}^N = (v^{N, j})_{1 \leq j \leq N}\) being a solution to 16 .

We are now ready to state our convergence result via regime I. Given \(N \in \mathbb{N}\), a function space \(X\), and \(\mathbf{f} = (f^{j})_{1 \leq j \leq N} \in X^{\otimes N}\), we define the norm \[\begin{align} \| \mathbf{f} \|_{\mathcal{A}_N X} = \| f^{j} \|_{\mathcal{A}_{N, j} X} \stackrel{\mathrm{def}}{=}\bigg( \frac{1}{N} \sum_{j = 1}^N \| f^{j} \|_X^2 \bigg)^{\frac{1}{2}} . \label{defANX} \end{align}\tag{19}\]

For part (i) of Theorem 3, we note that pathwise global well-posedness for the \(\mathrm{PLSM}_N\) was established in [4], but, once again, in a different formulation from that of ours. In [4], the initial condition is \(\mathbf{u}_0^N \in L^2 (\mathbb{T}^2)^{\otimes N}\) and each component \(v^{N, j}\) of the solution \(\mathbf{v}^N\) to the perturbed \(\mathrm{PLSM}_N\) 16 exists in the space \[\begin{align} C ([0, T]; L^2 (\mathbb{T}^2)) \cap L^4 ([0, T]; L^4 (\mathbb{T}^2)) \cap L^2 ([0, T]; H^1 (\mathbb{T}^2)) \end{align}\]

for any \(T > 0\). In this paper, we need higher regularity of the solution to the perturbed \(\mathrm{PLSM}_N\) 16 , and so we present the additional treatment in Subsection 3.1.

Part (iii) of Theorem 3, the convergence from the \(\mathrm{PLSM}_N\) to the mean-field SNLH, has already been established in [4].¹ In fact, by using the first order expansions 17 and 14 , the authors in [4] showed that each component \(v^{N, j}\) of the solution \(\mathbf{v}^N\) to the perturbed \(\mathrm{PLSM}_N\) 16 converges in probability to \(v^j\) satisfying the perturbed mean-field SNLH 12 in the space \(C([0, T]; L^2 (\mathbb{T}^2))\) for any \(T > 0\). Moreover, under an additional assumption that the initial data are exchangeable (i.e. all permutations of the sequence of initial data have the same joint probability distribution), the authors in [4] established a convergence result in the \(L^2 (\Omega)\) sense. We refer the readers to [4] for more details.

Part (ii) of Theorem 3 is precisely the Smoluchowski-Kramers approximation for the hyperbolic \(O(N)\) linear sigma model. When \(N = 1\), this corresponds to the Smoluchowski-Kramers approximation for the cubic SdNLW studied in [37]. We remark that the convergence result obtained in [37] holds locally in time, whereas in this paper, we establish global-in-time convergence of coupled SdNLW systems by using a continuity argument (see Remark 4 below for a further comment). See Section 4 for details.

We now look at regime II. For the equation 7 , we fix \(0 < \varepsilon\leq 1\) and take \(N \to \infty\) to formally obtain the following equation: \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) v_\varepsilon^j = - 2 \mathbb{E}[\Psi_\varepsilon^j v_\varepsilon^j] \Psi_\varepsilon^j - 2 \mathbb{E}[\Psi_\varepsilon^j v_\varepsilon^j] v_\varepsilon^j - \mathbb{E}[(v_\varepsilon^j)^2] \Psi_\varepsilon^j - \mathbb{E}[(v_\varepsilon^j)^2] v_\varepsilon^j . \label{NLWev} \end{align}\tag{20}\]

By writing \[\begin{align} u_\varepsilon^j = \Psi_\varepsilon^j + v_\varepsilon^j, \label{expe} \end{align}\tag{21}\]

we can write out the following renormalized version of the mean-field \(\mathrm{SdNLW}_\varepsilon\) 4 : \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^j = - \mathbb{E}[ (u_\varepsilon^j)^2 - (\Psi_\varepsilon^j)^2 ] u_\varepsilon^j + \sqrt{2} \xi^j . \label{NLWeu} \end{align}\tag{22}\]

Again, we say that \(u_\varepsilon^j\) is a solution to the renormalized mean-field \(\mathrm{SdNLW}_\varepsilon\) 22 if \(u_\varepsilon^j\) is given by 21 with \(v_\varepsilon^j\) being a solution to the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) 20 .

We are now ready to state our convergence result via regime II.

The proof of part (i) of Theorem 5, global well-posedness for the mean-field \(\text{SdNLW}_\varepsilon\) 22 , follows from minor modifications of the proof for [1]. Similar to Proposition 1 on pathwise global well-posedness for \(\text{HLSM}_{\varepsilon, N}\), we can first construct a unique local-in-time solution to the perturbed mean-field \(\text{SdNLW}_\varepsilon\) 20 in \(L^2 (\Omega; C([0, T]; \mathcal{H}^s (\mathbb{T}^2)))\) for some \(T > 0\) by using the one degree of smoothing from the linear damped wave dynamics from 28 . Then, we can prove an a priori bound for the solution and extend it to an arbitrarily large time interval by studying the following energy functional: \[\begin{align} E_\varepsilon(w, \partial_tw) \stackrel{\mathrm{def}}{=}\frac{1}{2} \mathbb{E}\bigg[ \int_{\mathbb{T}^2} \big( (\varepsilon\partial_tw)^2 + w^2 + |\nabla w|^2 \big) dx \bigg] + \frac{1}{4} \int_{\mathbb{T}^2} \big( \mathbb{E}[w^2] \big)^2 dx , \label{Eew} \end{align}\tag{23}\]

which is conserved under the flow of the mean-field NLW: \[\begin{align} (\varepsilon^2 \partial_t^2 + 1 - \Delta) w = - \mathbb{E}[w^2] w . \end{align}\]

Once again, we need to use the hybrid approach from [15] combining the \(I\)-method [43] and a Gronwall-type argument [44]. While one can refer to [1] for all the above steps that lead to global well-posedness for the mean-field \(\text{SdNLW}_\varepsilon\) 22 , we establish a stronger a priori bound for the solution to the perturbed mean-field \(\text{SdNLW}_\varepsilon\) 20 in this paper. The main novelty here is that we are able to obtain an a priori bound uniformly in \(0 < \varepsilon\leq 1\). Such a uniform a priori bound plays a central role in proving convergence of the dynamics in part (iii) of Theorem 5 to be mentioned below.

Part (ii) of Theorem 5, the mean-field convergence of \(\text{HLSM}_{\varepsilon, N}\), has essentially been established in [1]. To prove the mean-field convergence, we can establish law-of-large-numbers type lemmas using the second moment bound on the solution \(v_\varepsilon^j\) to the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) 20 . Since \(\varepsilon\) is fixed, we do not need to worry about dependence on \(\varepsilon\), and so we can proceed in exactly the same manner as the proof of [1]. We omit details.

Part (iii) of Theorem 5 corresponds to the Smoluchowski-Kramers approximation for the mean-field SdNLW. In contrast to the pathwise setting in part (ii) of Theorem 3, the solution to the mean-field equations are in the \(L^2 (\Omega)\) sense. This means that the continuity argument based on a pathwise solution theory used for proving the convergence via regime I is no longer suitable in proving part (iii) of Theorem 5. Nevertheless, as mentioned above, we can establish a uniform-in-\(\varepsilon\) bound for the solution to the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) 20 ; see Proposition 14. This uniform bound allows us to prove global-in-time convergence of the dynamics in a direct manner. Let us remark that in establishing the uniform-in-\(\varepsilon\) bound, we need to rely heavily on the fact that we are working with the mean-field nonlinearity. This is relevant in Lemma 13 below. One can still prove a bound as in Lemma 13 with the usual cubic nonlinearity but with more loss, which seems to be unacceptable in applying the Gronwall-type argument. See Section 5 for more details. This also explains why we use two entirely different approaches in proving the two Smoluchowski-Kramers approximation results in Theorem 3 (ii) and Theorem 5 (iii).

Before closing the introduction, we would like to remark that the coefficient \(\sqrt{2}\) in front of the space-time white noises in the models 1 , 2 , 3 , and 4 does not play any role in the analysis, and one can replace it by any other constants. However, if one consider Gibbs measures and invariant Gibbs dynamics, then it is crucial to have the coefficient \(\sqrt{2}\). We include a discussion on invariant Gibbs dynamics in Appendix 6, where we show the commutative diagram in Figure 1 at Gibbs equilibrium in Theorem 17. The proof follows essentially from same steps for proving Theorem 3 and Theorem 5 with only a few additional ingredients.

We end the introduction by stating several remarks.

2 Function spaces and preliminary lemmas↩︎

In this section, we introduce notations and recall some useful tools that will be applied in this paper.

We denote by \(C\) a constant that may vary from line to line. We also write \(C (\cdot)\) to emphasize dependence of this constant on external parameters, such as \(\omega\), \(N\), and \(T\). For two positive quantities \(a\) and \(b\), we write \(a \lesssim b\) if \(a \leq Cb\) for some constant \(C > 0\) that is independent of the set where \(a\) and \(b\) are allowed to vary. We write \(a \sim b\) if \(a \lesssim b\) and \(b \lesssim a\). We also use subscripts such as “\(\lesssim_{s, t}\)” to denote dependence on external parameters.

For any function \(f\) on \(\mathbb{T}^2\), we denote by \(\widehat f\) or \(\mathcal{F}(f)\) the Fourier transform of \(f\). For any function \(g\) on \(\mathbb{Z}^2\), we denote by \(g^\vee\) the inverse Fourier transform of \(g\). We will frequently use the Japanese bracket \(\langle \cdot \rangle = (1 + |\cdot|^2)^{\frac{1}{2}}\).

For function spaces, we often use shorthand notations, such as \(C_I H^s_x = C (I; H^s (\mathbb{T}^2))\) for some time interval \(I \subset \mathbb{R}\) and \(s \in \mathbb{R}\). If \(I = [0, T]\) for some \(T > 0\), we write \(C_T H_x^s = C_{[0, T]} H_x^s\). Given a Banach space \(X\) and \(N \in \mathbb{N}\), we define the \(\mathcal{A}_N X\)-norm of \(\mathbf{f} = (f^{j})_{1 \leq j \leq N} \in X^{\otimes N}\) as in 19 , namely the \(\ell^2\)-average of the \(X\)-norms of \(f^{j}\)’s. Similarly, we define the \(\mathcal{A}_N^{(2)} X\)-norm of \(\mathbf{g} = (g^{j, k})_{1 \leq j, k \leq N} \in X^{\otimes N^2}\) by \[\begin{align} \| \mathbf{g} \|_{\mathcal{A}_N^{(2)} X} = \big\| (g^{j, k})_{1 \leq j, k \leq N} \big\|_{\mathcal{A}_{N, j, k}^{(2)} X} \stackrel{\mathrm{def}}{=} \big\| (g^{j, k})_{1 \leq j, k \leq N} \big\|_{\mathcal{A}_{N, j} \mathcal{A}_{N, k} X} = \bigg( \frac{1}{N^2} \sum_{j, k = 1}^N \| g^{j, k} \|_X^2 \bigg)^{\frac{1}{2}} . \label{defAN2X} \end{align}\tag{24}\]

2.1 Sobolev and Besov spaces↩︎

Given \(s \in \mathbb{R}\) and \(1 \leq p \leq \infty\), we denote by \(W^{s, p} (\mathbb{T}^2)\) the \(L^p\)-based Sobolev space via the norm \[\begin{align} \| f \|_{W^{s, p}} \stackrel{\mathrm{def}}{=}\| \langle \nabla \rangle^s f \|_{L^p} = \| ( \langle \cdot \rangle^s \widehat f )^\vee \|_{L^p} . \end{align}\]

When \(p = 2\), we write \(H^s (\mathbb{T}^2) = W^{s, 2} (\mathbb{T}^2)\). By Plancherel’s identity, we have \[\begin{align} \| f \|_{H^s} = \| \langle \nabla \rangle^s f \|_{L^2} = \| \langle \cdot \rangle^s \widehat f (\cdot) \|_{\ell^2} . \end{align}\]

We also define the product space \(\mathcal{H}^s (\mathbb{T}^2) \stackrel{\mathrm{def}}{=}H^s (\mathbb{T}^2) \times H^{s - 1} (\mathbb{T}^2)\) via the norm \[\begin{align} \| (f, g) \|_{\mathcal{H}^s} \stackrel{\mathrm{def}}{=}\| f \|_{H^s} + \| g \|_{H^{s - 1}} . \end{align}\]

For any \(s_1 \leq s_2\) and \(1 \leq p_1 \leq p_2 \leq \infty\), we have the embedding \[\begin{align} \| f \|_{W^{s_1, p_1}} \lesssim\| f \|_{W^{s_2, p_2}} , \end{align}\]

which we will use frequently without mentioning.

We record the following product estimates for Sobolev spaces. See [14] and also [49] for the endpoint case.

We also need the Besov-Hölder spaces. Let \(\varphi : \mathbb{R}\to [0, 1]\) be a smooth even cutoff function supported on \([- \frac{8}{5}, \frac{8}{5}]\) such that \(\varphi \equiv 1\) on \([- \frac{5}{4}, \frac{5}{4}]\). Given \(\xi \in \mathbb{R}^2\), we set \(\varphi_1 (\xi) = \varphi (|\xi|)\) and, given a dyadic number \(N \geq 2\), \[\begin{align} \varphi_N (\xi) = \varphi (\tfrac{|\xi|}{N}) - \varphi (\tfrac{2 |\xi|}{N}). \end{align}\]

Given dyadic \(N \geq 1\), we define the Littlewood-Paley projector \(\mathbf{P}_N\) as the Fourier multiplier operator with symbol \(\varphi_N\). Given \(s \in \mathbb{R}\), we denote by \(\mathcal{C}^s (\mathbb{T}^2)\) the Besov-Hölder space via the norm \[\begin{align} \| f \|_{\mathcal{C}^s} \stackrel{\mathrm{def}}{=}\sup_{\substack{N \geq 1 \\ \text{dyadic}}} N^s \| \mathbf{P}_N f \|_{L^\infty} . \end{align}\]

We have the following embedding results between the Besov-Hölder spaces and the \(L^p\)-based Sobolev spaces. For a proof, see [50].

We also record the following product estimates for Besov-Hölder spaces. For a proof, see, for example, [50].

We now record the following estimate on the smoothing effect of the heat semigroup. The proof follows from minor modifications of [51].

2.2 Convergence of linear propagators↩︎

In this subsection, we recall some known results on the convergence of the linear propagators of the damped wave equation and the heat equation. We follow the notations in [37].

Given \(0 < \varepsilon\leq 1\), we define \(\mathbf{P}_\varepsilon^{\text{lo}}\) and \(\mathbf{P}_\varepsilon^{\text{hi}}\) as the sharp frequency projections onto \(\{ n \in \mathbb{Z}^2: \langle n \rangle \leq (2 \varepsilon)^{-1} \}\) and \(\{ n \in \mathbb{Z}^2 : \langle n \rangle > (2 \varepsilon)^{-1} \}\), respectively. For any \(n \in \mathbb{Z}^2\), we also define \[\begin{align} \lambda_\varepsilon(n) \stackrel{\mathrm{def}}{=}\frac{\sqrt{1 - 4 \varepsilon^2 \langle n \rangle^2}}{2 \varepsilon^2} \quad \text{and} \quad \zeta_\varepsilon(n) \stackrel{\mathrm{def}}{=}\frac{\sqrt{4 \varepsilon^2 \langle n \rangle^2 - 1}}{2 \varepsilon^2} . \end{align}\]

We then define \[\begin{align} \mathcal{D}_\varepsilon(t) \stackrel{\mathrm{def}}{=}e^{- \frac{t}{2 \varepsilon^2}} \frac{\sinh ( t \lambda_\varepsilon(\nabla))}{\lambda_\varepsilon(\nabla)} \mathbf{P}_\varepsilon^{\text{lo}} + e^{- \frac{t}{2 \varepsilon^2}} \frac{\sin (t \zeta_\varepsilon(\nabla))}{\zeta_\varepsilon(\nabla)} \mathbf{P}_\varepsilon^{\text{hi}} , \label{defDe} \end{align}\tag{25}\]

whose associated Fourier multiplier is given by \[\begin{align} \widehat{\mathcal{D}}_\varepsilon(t, n) \stackrel{\mathrm{def}}{=}e^{- \frac{t}{2 \varepsilon^2}} \frac{\sinh ( t \lambda_\varepsilon(n))}{\lambda_\varepsilon(n)} \mathbf{1}_{\{ \langle n \rangle \leq (2 \varepsilon)^{-1} \}} + e^{- \frac{t}{2 \varepsilon^2}} \frac{\sin (t \zeta_\varepsilon(n))}{\zeta_\varepsilon(n)} \mathbf{1}_{\{ \langle n \rangle > (2 \varepsilon)^{-1} \}} . \label{Den} \end{align}\tag{26}\]

Given \(0 < \varepsilon\leq 1\), the solution to the inhomogeneous linear damped wave equation \[\begin{align} \begin{cases} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u = F \\ (u, \partial_tu)|_{t = 0} = (\phi_0, \phi_1) \end{cases} \end{align}\]

is given by \[\begin{align} u (t) = P_\varepsilon(t) (\phi_0, \phi_1) + \mathcal{I}_\varepsilon(F) (t) , \end{align}\]

where \[\begin{align} P_\varepsilon(t) (\phi_0, \phi_1) \stackrel{\mathrm{def}}{=}(\varepsilon^{-2} + \partial_t) \mathcal{D}_\varepsilon(t) \phi_0 + \mathcal{D}_\varepsilon(t) \phi_1 \label{defPe} \end{align}\tag{27}\]

and \[\begin{align} \mathcal{I}_{\varepsilon} (F) (t) \stackrel{\mathrm{def}}{=}\int_{0}^t \varepsilon^{-2} \mathcal{D}_\varepsilon(t - t') F (t') dt' . \label{defIe} \end{align}\tag{28}\]

We also consider the linear heat equation \[\begin{align} \begin{cases} (\partial_t+ 1 - \Delta) u = F \\ u|_{t = 0} = \phi_0 , \end{cases} \end{align}\]

whose solution is given by \[\begin{align} u(t) = P_0 (t) \phi_0 + \mathcal{I}_0 (F) (t) , \end{align}\]

where \[\begin{align} P_0 (t) \phi_0 \stackrel{\mathrm{def}}{=}e^{- t (1 - \Delta)} \phi_0 \label{defP0} \end{align}\tag{29}\]

and \[\begin{align} \mathcal{I}_{0} (F) (t) \stackrel{\mathrm{def}}{=}\int_{0}^t P_0 (t - t') F (t') dt' . \label{defI0} \end{align}\tag{30}\]

For later convenience, we alse write \[\begin{align} P_0 (t) (\phi_0, \phi_1) = P_0 (t) \phi_0 . \label{P001} \end{align}\tag{31}\]

The following lemma was proved in [37].

2.3 \(I\)-method and \(I\)-operator↩︎

As mentioned in the introduction, we will need the \(I\)-method to establish a global-in-time a priori bound for the solution of the mean-field \(\text{SdNLW}_{\varepsilon}\).

Let \(0 < s < 1\). Given \(M \in \mathbb{N}\), we define a smooth, radial, and non-increasing (in radial direction) function \(m_{s, M} \in C^\infty (\mathbb{R}^2; [0, 1])\) satisfying \[\begin{align} m_{s, M} (\xi) = \begin{cases} 1 & \text{if } |\xi| \leq M \\ (\frac{M}{|\xi|})^{1 - s} & \text{if } |\xi| \geq 2 M . \end{cases} \end{align}\]

We then define \(I_M\) (omitting the dependence on \(s\)) as the Fourier multiplier operator with symbol \(m_{s, M}\): \[\begin{align} \widehat{I_M f} (n) = m_{s, M} (n) \widehat f (n) \label{defIop} \end{align}\tag{32}\]

for any \(n \in \mathbb{Z}^2\). Note that we can easily deduce from the definition that for any \(\gamma \in \mathbb{R}\), \[\begin{align} \| f \|_{H^{\gamma}} \lesssim\| I_M f \|_{H^{\gamma + 1 - s}} \lesssim M^{1 - s} \| f \|_{H^\gamma} . \label{Ibdd1} \end{align}\tag{33}\]

Also, by using the Littlewood-Paley theorem, for any \(s_0 \in \mathbb{R}\), \(0 \leq s_1 \leq 1 - s\), and \(1 < p < \infty\), we have \[\begin{align} \| I_M f \|_{W^{s_0 + s_1, p}} \lesssim M^{s_1} \| f \|_{W^{s_0, p}} . \label{Ibdd2} \end{align}\tag{34}\]

Let us record some useful estimates for the \(I\)-operator. The following lemma was proved in [15].

We also have the following lemma, whose proof follows from [15].

2.4 On the stochastic convolution↩︎

In this subsection, we consider the stochastic convolutions \(\Psi_\varepsilon^j\) and \(\Psi_0^j\) given by 5 and 13 , respectively.

We recall that \(\{\xi^j\}_{j \in \mathbb{N}}\) is a family of independent space-time white noises on \(\mathbb{R}_+ \times \mathbb{T}^2\). From 5 , given any \(0 < \varepsilon\leq 1\), we may write \[\begin{align} \Psi_\varepsilon^j (t) = \sqrt{2} \int_0^t \varepsilon^{-2} \mathcal{D}_\varepsilon(t - t') d W^j (t') , \end{align}\]

where \(\mathcal{D}_\varepsilon(t)\) is defined in 25 and \[\begin{align} W^j (t) \stackrel{\mathrm{def}}{=}\frac{1}{2 \pi} \sum_{n \in \mathbb{Z}^2} B_n^j (t) e^{i n \cdot x} \label{Wj} \end{align}\tag{35}\]

is a cylindrical Wiener process on \(L^2 (\mathbb{T}^2)\) with \(B_n^j (t) = (2 \pi)^{-1} \langle \xi^j, \mathbf{1}_{[0, t]} (t') e^{i n \cdot x} \rangle_{t', x}\), where \(\langle \cdot, \cdot \rangle_{t, x}\) denotes the duality pairing on \(\mathbb{R}_+ \times \mathbb{T}^2\). Note that the definition implies that \(B_0^j\) is a standard real-valued Brownian motion and \(\{ B_n^j \}_{n \in \mathbb{Z}^2 \setminus \{0\}}\) is a family of independent standard complex-valued Brownian motions conditioned such that \(B_{- n}^j = \overline{B_n^j}\) for any \(n \in \mathbb{Z}^2 \setminus \{0\}\). When \(\varepsilon= 0\), from 13 , we write \[\begin{align} \Psi_0^j (t) = \sqrt{2} \int_0^t P_0 (t - t') d W^j (t') , \end{align}\]

where \(P_0\) is defined in 29 .

Given \(M \in \mathbb{N}\), let \(P_{\leq M}\) be the sharp frequency projection onto \(\{ n \in \mathbb{Z}^2 : |n| \leq M \}\). Given \(M \in \mathbb{N}\), \(j \in \mathbb{N}\), and \(0 \leq \varepsilon\leq 1\), we define \(\Psi_{\varepsilon, M}^j \stackrel{\mathrm{def}}{=}P_{\leq M} \Psi_\varepsilon^j\). Note that for each \(n \in \mathbb{Z}^2\), we have from [37] that \[\begin{align} \mathbb{E}\Big[ | \widehat{\Psi_\varepsilon^j} (t, n) |^2 \Big] \lesssim\langle n \rangle^{-2} \label{Psin} \end{align}\tag{36}\]

with the underlying constant independent of \(\varepsilon\) and \(t\). Also, a direct computation yields that when \(0 < \varepsilon\leq 1\), we have \[\begin{align} \mathbb{E}\Big[ | \widehat{\Psi_\varepsilon^j} (t, n) |^2 \Big] &\geq \mathbf{1}_{\{ \langle n \rangle > (2 \varepsilon)^{-1} \}} \Big( \frac{1}{\langle n \rangle^2} - e^{- \frac{t}{\varepsilon^2}} \frac{4 \varepsilon^2}{4 \varepsilon^2 \langle n \rangle^2 - 1} \\ &\quad - e^{- \frac{t}{\varepsilon^2}} \frac{\sin (2 t \zeta_\varepsilon(n))}{\langle n \rangle^2 \sqrt{4 \varepsilon^2 \langle n \rangle^2 - 1}} + e^{- \frac{t}{\varepsilon^2}} \frac{\cos (2 t \zeta_\varepsilon(n))}{\langle n \rangle^2 (4 \varepsilon^2 \langle n \rangle^2 - 1)} \Big) \\ &\gtrsim_{\varepsilon, t} \mathbf{1}_{\{ \langle n \rangle \geq \varepsilon^{-1} \}} \frac{1}{\langle n \rangle^2} \end{align}\]

at least for \(t \gtrsim\varepsilon^{2}\), and when \(\varepsilon= 0\), we have \[\begin{align} \mathbb{E}\Big[ |\widehat{\Psi_0^j} (t, n)|^2 \Big] = \frac{1 - e^{-2 t \langle n \rangle^2}}{\langle n \rangle^2} \gtrsim_t \frac{1}{\langle n \rangle^2} \end{align}\]

for \(t > 0\). Thus, we have \[\begin{align} \sigma_{\varepsilon, M} (t) &\stackrel{\mathrm{def}}{=}\mathbb{E}\big[ |\Psi_{\varepsilon, M}^j (t) |^2 \big] \sim_{\varepsilon, t} \log M , \label{sigmaM} \end{align}\tag{37}\]

which diverges as \(M \to \infty\). Given \(k, j \in \mathbb{N}\), we define the Wick products \[\begin{align} \begin{aligned} :\!{ \Psi_{\varepsilon, M}^k \Psi_{\varepsilon, M}^j }\!: &\stackrel{\mathrm{def}}{=} \begin{cases} H_2 ( \Psi_{\varepsilon, M}^j ; \sigma_{\varepsilon, M} ) & \text{if } k = j \\ \Psi_{\varepsilon, M}^k \Psi_{\varepsilon, M}^j & \text{if } k \neq j , \end{cases} \\ :\!{ (\Psi_{\varepsilon, M}^k)^2 \Psi_{\varepsilon, M}^j }\!: &\stackrel{\mathrm{def}}{=} \begin{cases} H_3 ( \Psi_{\varepsilon, M}^j ; \sigma_{\varepsilon, M} ) & \text{if } k = j \\ H_2 ( \Psi_{\varepsilon, M}^k ; \sigma_{\varepsilon, M} ) \Psi_{\varepsilon, M}^j & \text{if } k \neq j , \end{cases} \end{aligned} \label{wick0} \end{align}\tag{38}\]

where \(H_2 (\cdot ; \sigma)\) and \(H_3 (\cdot; \sigma)\) denote Hermite polynomials of degree 2 and 3, respectively, with variance parameter \(\sigma> 0\). Then, the Wick products appearing earlier in 7 are defined by \[\begin{align} \begin{aligned} :\!{ \Psi_{\varepsilon}^k \Psi_{\varepsilon}^j }\!: &\stackrel{\mathrm{def}}{=}\lim_{M \to \infty} :\!{ \Psi_{\varepsilon, M}^k \Psi_{\varepsilon, M}^j }\!: , \\ :\!{ (\Psi_{\varepsilon}^k)^2 \Psi_{\varepsilon}^j }\!: &\stackrel{\mathrm{def}}{=}\lim_{M \to \infty} :\!{ (\Psi_{\varepsilon, M}^k)^2 \Psi_{\varepsilon, M}^j }\!: . \end{aligned} \label{wick} \end{align}\tag{39}\]

We will see in Lemma 8 below that the limits in 39 exist almost surely in the space \(C(\mathbb{R}_+; W^{- \sigma, \infty})\) for any \(\sigma> 0\).

We now state and prove the following lemma regarding the regularity properties of the above stochastic objects.

We will also need the following lemma. Let us recall the \(I\)-operator in 32 .

3 Global well-posedness for the \(\mathrm{PLSM}_N\) and the mean-field SNLH↩︎

In this section, we prove global well-posedness of the \(\mathrm{PLSM}_N\) 18 as stated in Theorem 3 (i) and the mean-field SNLH 15 as stated in Proposition 2.

3.1 Global well-posedness for the \(\text{PLSM}_N\)↩︎

Let us consider the perturbed \(\mathrm{PLSM}_N\) 16 , whose Duhamel formulation is given by \[\begin{align} \begin{aligned} v^{N, j} (t) = P_0 (t) u_0^{N, j} - \frac{1}{N} &\sum_{k = 1}^N \mathcal{I}_0 \big( :\!{ (\Psi_0^k)^2 \Psi_0^j }\!: + :\!{ (\Psi_0^k)^2 }\!: v^{N, j} + 2 v^{N, k} :\!{ \Psi_0^k \Psi_0^j }\!: \\ &+ 2 \Psi_0^k v^{N, k} v^{N, j} + (v^{N, k})^2 \Psi_0^j + (v^{N, k})^2 v^{N, j} \big) (t) , \quad j = 1, \dots, N, \end{aligned} \label{NLHNvDuh} \end{align}\tag{40}\]

where \(P_0\) is defined in 29 , \(\mathcal{I}_0\) is defined in 30 , and \(\mathbf{u}_0^N = (u_0^{N, j})_{1 \leq j \leq N}\) is the initial data for 16 . The system 40 is known to be globally well-posed by [4], but does not satisfy the regularity requirement in our setting. Our goal is to prove the following proposition on global well-posedness of the system 40 , which implies Theorem 3 (i) on pathwise global well-posedness for the \(\mathrm{PLSM}_N\) 18 .

3.2 Global well-posedness for the mean-field SNLH↩︎

Let us consider the perturbed mean-field SNLH 12 and drop the superscript \(j\) in this subsection for simplicity. The Duhamel formulation of 12 is given by \[\begin{align} v (t) = P_0 (t) u_0 - \mathcal{I}_0 \big( 2 \mathbb{E}[\Psi_0 v] \Psi_0 + 2 \mathbb{E}[\Psi_0 v] v + \mathbb{E}[v^2] \Psi_0 + \mathbb{E}[v^2] v \big) (t) , \label{NLHvDuh} \end{align}\tag{58}\]

where \(P_0\) is defined in 29 , \(\mathcal{I}_0\) is defined in 30 , and \(u_0\) is the initial data for 12 . The equation 58 is known to be globally well-posed by [4], but we need better regularity of the solution. Our goal is to prove the following proposition on global well-posedness of the equation 58 , which implies Proposition 2 on global well-posedness for the mean-field SNLH 15 .

4 Convergence via regime I↩︎

In this section, we prove Theorem 3, the convergence of the \(\mathrm{HLSM}_{\varepsilon, N}\) 8 to the mean-field SNLH 15 via regime I.

We first note that part (i) of Theorem 3, pathwise global well-posedness for \(\mathrm{PLSM}_N\) 18 , has already been established via Proposition 10. Also, as mentioned in the introduction, part (iii) of Theorem 3, the convergence from \(\mathrm{PLSM}_N\) 18 to the mean-field SNLH 15 , has been covered by [4]. Thus, we focus on proving part (ii) of Theorem 3, the convergence of the \(\mathrm{HLSM}_{\varepsilon, N}\) 8 to the \(\mathrm{PLSM}_N\) 18 as \(\varepsilon\to 0\).

To prove the theorem, we need to show the convergence of \(\mathbf{v}_\varepsilon^N = (v_\varepsilon^{N, j})_{1 \leq j \leq N}\) satisfying the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) \[\begin{align} \begin{aligned} v_\varepsilon^{N, j} (t) &= P_\varepsilon(t) (u_0^{N, j}, u_1^{N, j}) - \frac{1}{N} \sum_{k = 1}^N \mathcal{I}_\varepsilon\big( :\!{ (\Psi_\varepsilon^k)^2 \Psi_\varepsilon^j }\!: + :\!{ (\Psi_\varepsilon^k)^2 }\!: v_\varepsilon^{N, j} \\ &\qquad + 2 v_\varepsilon^{N, k} :\!{ \Psi_\varepsilon^k \Psi_\varepsilon^j }\!: + 2 \Psi_\varepsilon^k v_\varepsilon^{N, k} v_\varepsilon^{N, j} + (v_\varepsilon^{N, k})^2 \Psi_\varepsilon^j + (v_\varepsilon^{N, k})^2 v_\varepsilon^{N, j} \big) (t) \end{aligned} \label{NLWNevDuh2} \end{align}\tag{66}\]

to \(\mathbf{v}^N = (v^{N, j})_{1 \leq j \leq N}\) satisfying the perturbed \(\mathrm{PLSM}_N\) \[\begin{align} \begin{aligned} v^{N, j} (t) &= P_0 (t) (u_0^{N, j}, u_1^{N, j}) - \frac{1}{N} \sum_{k = 1}^N \mathcal{I}_0 \big( :\!{ (\Psi_0^k)^2 \Psi_0^j }\!: + :\!{ (\Psi_0^k)^2 }\!: v^{N, j} \\ &\qquad + 2 v^{N, k} :\!{ \Psi_0^k \Psi_0^j }\!: + 2 \Psi_0^k v^{N, k} v^{N, j} + (v^{N, k})^2 \Psi_0^j + (v^{N, k})^2 v^{N, j} \big) (t) , \end{aligned} \label{NLHNvDuh2} \end{align}\tag{67}\]

where \(P_\varepsilon\) is defined in 27 , \(\mathcal{I}_\varepsilon\) is defined in 28 , \(P_0\) is defined in 29 (see also 31 ), \(\mathcal{I}_0\) is defined in 30 , and \((\mathbf{u}_0^N, \mathbf{u}_1^N) = ((u_0^{N, j}, u_1^{N, j}))_{1 \leq j \leq N}\) is the initial data for both systems. Our goal is to prove the following proposition, which, together with 6 , 17 , and the convergence of \(\Psi_\varepsilon^j\) to \(\Psi_0^j\) from Lemma 8 (iii), implies Theorem 3 (ii).

5 Convergence via regime II↩︎

In this section, we prove Theorem 5, convergence of \(\mathrm{HLSM}_{\varepsilon, N}\) 8 to the mean-field SNLH 15 via regime II.

As mentioned earlier, part (i) and part (ii) of Theorem 5, global well-posedness for the mean-field \(\mathrm{SdNLW}_\varepsilon\) and mean-field convergence of the \(\text{HLSM}_{\varepsilon, N}\), have been essentially established by [1] and [1], respectively. Thus, we focus on proving part (iii) of Theorem 5, convergence of the mean-field \(\mathrm{SdNLW}_\varepsilon\) 22 to the mean-field SNLH 15 . As mentioned in the introduction, we first establish a uniform-in-\(\varepsilon\) bound for the solution in Subsection 5.1 and then show the global-in-time convergence of the dynamics in Subsection 5.2.

5.1 Uniform a priori bound for the mean-field \(\text{SdNLW}_\varepsilon\)↩︎

In this section, we establish a uniform-in-\(\varepsilon\) a priori bound for \(v_\varepsilon^j\) satisfying the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) 20 . Let us drop the superscript \(j\) in this subsection for simplicity. The Duhamel formulation of 20 is given by \[\begin{align} v_\varepsilon(t) = P_\varepsilon(t) (u_0, u_1) - \mathcal{I}_\varepsilon\big( 2 \mathbb{E}[ \Psi_\varepsilon v_\varepsilon] \Psi_\varepsilon+ 2 \mathbb{E}[\Psi_\varepsilon v_\varepsilon] v_\varepsilon+ \mathbb{E}[(v_\varepsilon)^2] \Psi_\varepsilon+ \mathbb{E}[(v_\varepsilon)^2] v_\varepsilon\big) (t) , \label{NLWveDuh} \end{align}\tag{89}\]

where \(P_\varepsilon\) is defined in 27 , \(\mathcal{I}_\varepsilon\) is defined in 28 , and \((u_0, u_1)\) is the initial data for 20 . Our goal is to prove the following global-in-time a priori bound for \(v_\varepsilon\) uniformly in \(\varepsilon\).

We recall the \(I_M\)-operator defined in 32 . Recalling the energy functional \(E_\varepsilon\) in 23 , in view of 33 , we see that it suffices to provide an upper bound for the following energy: \[\begin{align} \begin{aligned} E_{\varepsilon, M} (t) &= E_\varepsilon( I_M v_\varepsilon(t), \partial_tI_M v_\varepsilon(t) ) \\ &=\frac{1}{2} \mathbb{E}\bigg[ \int_{\mathbb{T}^2} (\varepsilon\partial_tI_M v_\varepsilon(t))^2 + (I_M v_\varepsilon(t))^2 + |\nabla I_M v_\varepsilon(t) |^2 dx \bigg] \\ &\quad + \frac{1}{4} \int_{\mathbb{T}^2} \mathbb{E}[ (I_M v_\varepsilon(t))^2 ]^2 dx . \end{aligned} \label{defEe} \end{align}\tag{90}\]

From 20 , given \(0 \leq t_0 \leq t \leq T\), we compute that \[\begin{align} \begin{aligned} E_{\varepsilon, M} &(t) - E_{\varepsilon, M} (t_0) + \mathbb{E}\bigg[ \int_{t_0}^t \int_{\mathbb{T}^2} (\partial_tI_M v_\varepsilon)^2 dx dt' \bigg] \\ &= \mathbb{E}\bigg[ \int_{t_0}^t \int_{\mathbb{T}^2} \partial_tI_M v_\varepsilon\Big( \mathbb{E}[ (I_M v_\varepsilon)^2 ] I_M v_\varepsilon- I_M \big( \mathbb{E}[v_\varepsilon^2] v_\varepsilon\big) \Big) dx dt' \bigg] \\ &\quad - \mathbb{E}\bigg[ \int_{t_0}^t \int_{\mathbb{T}^2} \partial_tI_M v_\varepsilon\Big( I_M \big( \mathbb{E}[v_\varepsilon^2] \Psi_\varepsilon\big) + 2 I_M \big( \mathbb{E}[\Psi_\varepsilon v_\varepsilon] v_\varepsilon\big) \Big) dx dt' \bigg] \\ &\quad - 2 \mathbb{E}\bigg[ \int_{t_0}^t \int_{\mathbb{T}^2} (\partial_tI_M v_\varepsilon) I_M \big( \mathbb{E}[ \Psi_\varepsilon v_\varepsilon] \Psi_\varepsilon\big) dx dt' \bigg] \\ &\stackrel{\mathrm{def}}{=}A_1 - A_2 - A_3 . \end{aligned} \label{defB123} \end{align}\tag{91}\]

We also define \[\begin{align} A_4 \stackrel{\mathrm{def}}{=}\mathbb{E}\bigg[ \int_{t_0}^t \int_{\mathbb{T}^2} \partial_tI_M v_\varepsilon\Big( \mathbb{E}[ (I_M v_\varepsilon)^2 ] I_M \Psi_\varepsilon+ 2 \mathbb{E}[ I_M \Psi_\varepsilon I_M v_\varepsilon] I_M v_\varepsilon\Big) dx dt' \bigg] . \label{defB4} \end{align}\tag{92}\]

In the following series of lemmas, we estimate the terms \(A_1\), \(A_2\), \(A_3\), and \(A_4\).

In the following lemma, we rely crucially on the fact that we are working with the mean-field nonlinearity. Specifically, we ensure a loss not worse than \(\log M\) in our estimates. If the bound were to grow faster than \(\log M\), the subsequent Gronwall-type argument would fail, yielding a singular estimate. Fortunately, the mean-field nonlinearity guarantees that the norm of \(I_M \Psi_\varepsilon\) remains well-behaved, growing at most like \(\log M\). One can compare this situation to [15], where the authors also ensure a loss of \(\log M\) when working with the usual cubic nonlinearity; however, in our setting, we must apply an additional Cauchy-Schwarz inequality in order to guarantee uniformity in \(\varepsilon\). Because of this additional estimate, we are unable to obtain a \(\log M\) bound in the \(N\)-component setting. Consequently, for the \(N\)-component system, the subsequent proof breaks down, preventing us from finding a uniform-in-\(\varepsilon\) bound for the solution.

Gathering 91 , Lemma 10, Lemma 11, Lemma 12, and Lemma 13, we obtain that given \(0 \leq t_0 \leq t \leq T\) with \(t - t_0 \leq 1\), \[\begin{align} \begin{aligned} &E_{\varepsilon, M} (t) - E_{\varepsilon, M} (t_0) \\ &\leq C(s) M^{- 6 s + 4} \int_{t_0}^t E_{\varepsilon, M} (t')^3 dt' + C (\nu, s) M^{- 2 s + 1 + 2 \nu} \int_{t_0}^t E_{\varepsilon, M} (t')^2 dt' \\ &\quad + C (s) \log M \int_{t_0}^t E_{\varepsilon, M} (t') dt' + C (\gamma) M^{2 \gamma} \int_{t_0}^t E_{\varepsilon, M} (t')^{1 - \frac{s}{2} + \frac{\gamma}{2}} dt' \end{aligned} \label{Ediff} \end{align}\tag{98}\]

for any \(\frac{2}{3} \leq s < 1\), \(M \in \mathbb{N}\) with \(M \geq 10\), \(0 < \nu \leq 1 - s\), and \(0 < \gamma \leq 1 - s\) and some constants \(C (s) > 0\), \(C(\nu, s) > 0\), \(C(\gamma) > 0\) independent of \(\varepsilon\).

Let us now show the following iterative lemma, for which the idea goes back to [15], [46].

We are now ready to prove the a priori bound in Proposition 14. The proof is similar to that in [15], [46], but for completeness we write out details.

5.2 Proof of the convergence↩︎

In this subsection, we prove part (iii) in Theorem 5, convergence of the mean-field \(\mathrm{SdNLW}_\varepsilon\) 22 to the mean-field SNLH 15 . As in the previous subsection, we drop the superscript \(j\). To prove it, we need to show the convergence of \(v_\varepsilon\) satisfying \[\begin{align} v_\varepsilon(t) = P_\varepsilon(t) (u_0, u_1) - \mathcal{I}_\varepsilon\big( 2 \mathbb{E}[ \Psi_\varepsilon v_\varepsilon] \Psi_\varepsilon+ 2 \mathbb{E}[\Psi_\varepsilon v_\varepsilon] v_\varepsilon+ \mathbb{E}[(v_\varepsilon)^2] \Psi_\varepsilon+ \mathbb{E}[(v_\varepsilon)^2] v_\varepsilon\big) (t) \label{NLWveDuh2} \end{align}\tag{114}\]

to \(v\) satisfying \[\begin{align} v (t) = P_0 (t) (u_0, u_1) - \mathcal{I}_0 \big( 2 \mathbb{E}[ \Psi_0 v ] \Psi_0 + 2 \mathbb{E}[\Psi_0 v] v + \mathbb{E}[v^2] \Psi_0 + \mathbb{E}[v^2] v \big) (t) , \label{NLHvDuh2} \end{align}\tag{115}\]

where \(P_\varepsilon\) is defined in 27 , \(\mathcal{I}_\varepsilon\) is defined in 28 , \(P_0\) is defined in 29 (see also 31 ), \(\mathcal{I}_0\) is defined in 30 , and \((u_0, u_1)\) is the initial data for both equations. Our goal is to prove the following proposition, which, together with 21 , 14 , and the convergence \(\Psi_\varepsilon\) to \(\Psi_0\) from Lemma 8, implies Theorem 5 (iii).

6 Smoluchowski-Kramers approximation for invariant Gibbs dynamics↩︎

In the appendix, we consider the commutative diagram in Figure 1 for global-in-time dynamics at Gibbs equilibrium.

6.1 Gibbs measures↩︎

Let us first keep our discussion at a formal level (i.e. we disregard the renormalization for now) and start with the Gibbs measure \(\vec{\rho}_\varepsilon^N\) for the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 , which is formally given by \[\begin{align} d \vec{\rho}_\varepsilon^N (\mathbf{u}^N, \partial_t\mathbf{u}^N) = Z_{\varepsilon, N}^{-1} e^{- E_\varepsilon^N (\mathbf{u}^N, \partial_t\mathbf{u}^N)} d \mathbf{u}^N d (\partial_t\mathbf{u}^N) , \end{align}\]

where \((\mathbf{u}^N, \partial_t\mathbf{u}^N) = ((u^{N, j}, \partial_tu^{N, j}))_{1 \leq j \leq N}\), \(E_\varepsilon^N\) is as defined in 9 , and \(Z_{\varepsilon, N}\) (along with all the “\(Z\)”’s below) is a normalizing factor. To be more precise, we write \(\vec{\rho}_\varepsilon^N\) as a weighted Gaussian measure as follows. Given \(s \in \mathbb{R}\) and \(0 < \varepsilon\leq 1\), we define the fractional Gaussian field \(\mu_{s, \varepsilon}\) with scaling parameter \(\varepsilon\) as \[\begin{align} d \mu_{s, \varepsilon} (u) = Z_{s, \varepsilon}^{-1} e^{- \frac{1}{2} \varepsilon^2 \| u \|_{H^s (\mathbb{T}^2)}^2} du \stackrel{\mathrm{def}}{=}\prod_{n \in \mathbb{Z}^2} Z_{s, \varepsilon, n}^{-1} e^{- \frac{1}{2} \varepsilon^2 \langle n \rangle^{2 s} |\widehat u (n)|^2} d \widehat u (n) . \label{defmu} \end{align}\tag{128}\]

From [57], we see that \(\mu_{s, \varepsilon}\) is a Gaussian probability measure on \(H^{s - 1 - \delta} (\mathbb{T}^2) \setminus H^{s - 1} (\mathbb{T}^2)\) for any \(\delta> 0\). When \(s = 1\) and \(\varepsilon= 1\), this measure is the massive Gaussian free field on \(\mathbb{T}^2\), in which case we write \(\mu_1 = \mu_{1, 1}\) for simplicity; when \(s = 0\) and \(\varepsilon= 1\), it is the white noise measure on \(\mathbb{T}^2\), in which case we write \(\mu_0 = \mu_{0, 1}\) for simplicity. Given a measure \(\mu\), we denote by \(\mu^{\otimes N}\) the \(N\)-fold product measure \(\mu \otimes \cdots \otimes \mu\). Then, defining the measure \[\begin{align} d \rho^N (\mathbf{u}^N) = Z_N^{-1} \exp \bigg( - \frac{1}{4N} \int_{\mathbb{T}^2} \Big( \sum_{j = 1}^N (u^{N, j})^2 \Big)^2 dx \bigg) d \mu_1^{\otimes N} (\mathbf{u}^N) , \label{rhoN0} \end{align}\tag{129}\]

we can write the joint measure as \[\begin{align} d \vec{\rho}_\varepsilon^N (\mathbf{u}^N, \partial_t\mathbf{u}^N) = d \rho^N (\mathbf{u}^N) \otimes d \mu_{0, \varepsilon}^{\otimes N} (\partial_t\mathbf{u}^N) . \label{rhoeN0} \end{align}\tag{130}\]

As mentioned in the introduction, the measure \(\rho^N\) is called the \(O(N)\) linear sigma model and was studied by Wilson in [10]. Using the notations from 11 , we can write the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 as \[\begin{align} \varepsilon^2 \partial_t\begin{pmatrix} \mathbf{u}^N \\ \partial_t\mathbf{u}^N \end{pmatrix} = \begin{pmatrix} 0 & \boldsymbol{Id}^N \\ - \boldsymbol{Id}^N & 0 \end{pmatrix} \nabla_{(L^2 (\mathbb{T}^2))^{\otimes 2N}} E_\varepsilon^N (\mathbf{u}^N, \partial_t\mathbf{u}^N) + \begin{pmatrix} 0 \\ - \partial_t\mathbf{u}^N + \sqrt{2} \pmb{\xi}^N \end{pmatrix} \end{align}\]

with \(\pmb{\xi}^N = (\xi^j)_{1 \leq j \leq N}\), which can be viewed as a superposition of the coupled NLW system 10 and an Ornstein-Uhlenbeck process (for the component \(\partial_t\mathbf{u}^N\)). In particular, the coupled NLW system 10 formally preserves the Gibbs measure \(\vec{\rho}_\varepsilon^N\) and the Ornstein-Uhlenbeck process (for the component \(\partial_t\mathbf{u}^N\)) preserves the scaled white noise measure \(\mu_{0, \varepsilon}^{\otimes N}\). Thus, in view of 130 , we see that the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 formally preserves the Gibbs measure \(\vec{\rho}_\varepsilon^N\). This invariance is also expected from the stochastic quantization point of view, as the \(\mathrm{HLSM}_{\varepsilon, N}\) 1 is the so-called canonical stochastic quantization or the Hamiltonian stochastic quantization for the Gibbs measure \(\vec{\rho}_\varepsilon^N\); see [13].

Let us now take into account the renormalization, which is required to make sense of the interaction potential in 129 due to the fact that the Gaussian free field \(\mu_1\) is only supported on distributions of negative regularity. For the \(O(N)\) linear sigma model \(\rho^N\), we have \[\begin{align} d \rho^N (\mathbf{u}^N) = Z_N^{-1} \exp \bigg( - \frac{1}{4N} \int_{\mathbb{T}^2} :\!{\Big( \sum_{j = 1}^N (u^{N, j})^2 \Big)^2}\!: dx \bigg) d \mu_1^{\otimes N} (\mathbf{u}^N) . \label{rhoN} \end{align}\tag{131}\]

Here, the renormalized interaction potential is defined by \[\begin{align} :\!{\Big( \sum_{j = 1}^N (u^{N, j})^2 \Big)^2}\!: \, = \sum_{k = 1}^N \sum_{j = 1}^N :\!{(u^{N, k})^2 (u^{N, j})^2}\!: \end{align}\]

with \[\begin{align} :\!{(u^{N, k})^2 (u^{N, j})^2}\!: \, = \begin{cases} :\!{(u^{N, j})^4}\!: & \text{if } k = j \\ :\!{(u^{N, k})^2}\!: :\!{(u^{N, j})^2}\!: & \text{if } k \neq j , \end{cases} \end{align}\]

where \(:\!{(u^{N, j})^2}\!:\) and \(:\!{(u^{N, j})^4}\!:\) denote the standard Wick powers. For a rigorous construction of \(\rho^N\), one can adapt the procedure from the scalar case (see [58]–[60]) to the vector-valued setting here (see also [4]). Then, from 130 and 131 , the renormalized Gibbs measure \(\vec{\rho}_\varepsilon^N\) is given by \[\begin{align} \begin{aligned} d \vec{\rho}_\varepsilon^N (\mathbf{u}^N, \partial_t\mathbf{u}^N) &= d \rho^N (\mathbf{u}^N) \otimes d \mu_{0, \varepsilon}^{\otimes N} (\partial_t\mathbf{u}^N) \\ &= Z_N^{-1} \exp \bigg( - \frac{1}{4N} \int_{\mathbb{T}^2} :\!{\Big( \sum_{j = 1}^N (u^{N, j})^2 \Big)^2}\!: dx \bigg) d \mu_1^{\otimes N} (\mathbf{u}^N) \otimes d \mu_{0, \varepsilon}^{\otimes N} (\partial_t\mathbf{u}^N) . \end{aligned} \label{rhoeN} \end{align}\tag{132}\]

6.2 On stochastic objects↩︎

Before moving on to the dynamical problem, we need to first discuss the solutions to the linear equations just as the stochastic convolutions \(\Psi_\varepsilon^j\) satisfying 5 and \(\Psi_0^j\) satisfying 13 . However, in the setting of Gibbs dynamics, we need to incorporate the random initial data.

Let \(\{ \phi_0^j, \phi_1^j \}_{j \in \mathbb{N}}\) be a set of independent Gaussian random distributions such that \[\begin{align} \mathop{\mathrm{Law}}(\phi_0^j) = \mu_1 \quad \text{and} \quad \mathop{\mathrm{Law}}(\phi_1^j) = \mu_0 \end{align}\]

for all \(j \in \mathbb{N}\), where \(\mu_1\) is the massive Gaussian free field on \(\mathbb{T}^2\) and \(\mu_0\) is the white noise measure on \(\mathbb{T}^2\). In fact, we can write \[\begin{align} \phi_0^j = \frac{1}{2 \pi} \sum_{n \in \mathbb{Z}^2} \frac{g_n^j}{\langle n \rangle} e^{in \cdot x} \quad \text{and} \quad \phi_1^j = \frac{1}{2 \pi} \sum_{n \in \mathbb{Z}^2} h_n^j e^{in \cdot x} , \label{rand95init} \end{align}\tag{133}\]

where \(\{ g_n^j, h_n^j \}_{n \in \mathbb{Z}^2, j \in \mathbb{N}}\) are independent standard complex-valued Gaussian random variables conditioned such that \(g_{-n}^j = \overline{g_n^j}\) and \(h_{-n}^j = \overline{h_n^j}\) for any \(n \in \mathbb{Z}^2\) and \(j \in \mathbb{N}\). By scaling, we note that given \(0 < \varepsilon\leq 1\), the family \(\{ \varepsilon^{-1} \phi_1^j \}_{j \in \mathbb{N}}\) satisfies \[\begin{align} \mathop{\mathrm{Law}}(\varepsilon^{-1} \phi_1^j) = \mu_{0, \varepsilon} \quad \text{for all } j \in \mathbb{N}, \end{align}\]

where the Gaussian measure \(\mu_{0, \varepsilon}\) is defined in 128 .

Given \(j \in \mathbb{N}\) and \(0 < \varepsilon\leq 1\), we define \(\Phi_\varepsilon^j\) as the stochastic object satisfying the following linear stochastic damped wave equation with a parameter \(\varepsilon\) and Gaussian initial data: \[\begin{align} \begin{cases} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) \Phi_\varepsilon^j = \sqrt{2} \xi^j \\ (\Phi_\varepsilon^j, \partial_t\Phi_\varepsilon^j) |_{t = 0} = (\phi_0^j, \varepsilon^{-1} \phi_1^j) . \end{cases} \label{Phije} \end{align}\tag{134}\]

Here, the family of Gaussian initial data \(\{ \phi_0^j, \phi_1^j \}_{j \in \mathbb{N}}\) is assumed to be independent of the family of space-time white noises \(\{ \xi^j \}_{j \in \mathbb{N}}\). When \(\varepsilon= 0\), we define \(\Phi_0^j\) as the stochastic convolution satisfying the following linear stochastic heat equation with Gaussian initial data: \[\begin{align} \begin{cases} (\partial_t+ 1 - \Delta) \Phi_0^j = \sqrt{2} \xi^j \\ \Phi_0^j |_{t = 0} = \phi_0^j . \end{cases} \label{Phij0} \end{align}\tag{135}\]

One can compare \(\Phi_\varepsilon^j\) with \(\Psi_\varepsilon^j\) satisfying 5 and \(\Phi_0^j\) with \(\Psi_0^j\) satisfying 13 .

From 134 and 135 , given \(0 < \varepsilon\leq 1\), we may write \[\begin{align} \Phi_\varepsilon^j (t) = P_\varepsilon(t) (\phi_0^j, \varepsilon^{-1} \phi_1^j) + \sqrt{2} \int_0^t \varepsilon^{-2} \mathcal{D}_\varepsilon(t - t') d W^j (t') \end{align}\]

and \[\begin{align} \Phi_0^j (t) = P_0 (t) \phi_0^j + \sqrt{2} \int_0^t P_0 (t - t') d W^j (t') , \end{align}\]

where \(P_\varepsilon(t)\) is defined in 27 , \(\mathcal{D}_\varepsilon(t)\) is defined in 25 , \(P_0 (t)\) is defined in 29 , and \(W^j\)’s are the cylindrical Wiener processes defined in 35 .

Given \(M \in \mathbb{N}\), \(j \in \mathbb{N}\), and \(0 \leq \varepsilon\leq 1\), we define \(\Phi_{\varepsilon, M}^j \stackrel{\mathrm{def}}{=}P_{\leq M} \Phi_{\varepsilon}^j\). Let us compute the variance of \(\Phi_{\varepsilon, M}^j\).

From Lemma 15, we have \[\begin{align} \alpha_M \stackrel{\mathrm{def}}{=}\mathbb{E}\big[ |\Phi_{\varepsilon, M}^j (t)|^2 \big] = \sum_{\substack{n \in \mathbb{Z}^2 \\ |n| \leq M}} \frac{1}{\langle n \rangle^2} \sim \log M , \end{align}\]

which diverges as \(M \to \infty\). We see that unlike \(\sigma_{\varepsilon, M}\) in 37 , \(\alpha_M\) is independent of \(\varepsilon\) and \(t\). This is due to the fact that the Gaussian measure \(\mu_1 \otimes \mu_{0, \varepsilon}\) (see 128 ) is invariant under the dynamics of the linear stochastic damped wave equation 134 for \(0 < \varepsilon\leq 1\) and \(\mu_1\) is invariant under the dynamics of the linear stochastic heat equation 135 for \(\varepsilon= 0\).

Similar to 38 , given \(k, j \in \mathbb{N}\), we define the Wick products \[\begin{align} \begin{aligned} :\!{ \Phi_{\varepsilon, M}^k \Phi_{\varepsilon, M}^j }\!: &\stackrel{\mathrm{def}}{=} \begin{cases} H_2 ( \Phi_{\varepsilon, M}^j ; \alpha_{M} ) & \text{if } k = j \\ \Phi_{\varepsilon, M}^k \Phi_{\varepsilon, M}^j & \text{if } k \neq j , \end{cases} \\ :\!{ (\Phi_{\varepsilon, M}^k)^2 \Phi_{\varepsilon, M}^j }\!: &\stackrel{\mathrm{def}}{=} \begin{cases} H_3 ( \Phi_{\varepsilon, M}^j ; \alpha_{M} ) & \text{if } k = j \\ H_2 ( \Phi_{\varepsilon, M}^k ; \alpha_{M} ) \Phi_{\varepsilon, M}^j & \text{if } k \neq j , \end{cases} \end{aligned} \end{align}\]

where \(H_2 (\cdot ; \sigma)\) and \(H_3 (\cdot; \sigma)\) denote Hermite polynomials of degree 2 and 3, respectively, with variance parameter \(\sigma> 0\). Then, we define \[\begin{align} \begin{aligned} :\!{ \Phi_{\varepsilon}^k \Phi_{\varepsilon}^j }\!: &\stackrel{\mathrm{def}}{=}\lim_{M \to \infty} :\!{ \Phi_{\varepsilon, M}^k \Phi_{\varepsilon, M}^j }\!: , \\ :\!{ (\Phi_{\varepsilon}^k)^2 \Phi_{\varepsilon}^j }\!: &\stackrel{\mathrm{def}}{=}\lim_{M \to \infty} :\!{ (\Phi_{\varepsilon, M}^k)^2 \Phi_{\varepsilon, M}^j }\!: . \end{aligned} \label{wick2} \end{align}\tag{136}\]

We will see in Lemma 17 below that the limits in 136 exist almost surely in the space \(C(\mathbb{R}_+; W^{- \sigma, \infty})\) for any \(\sigma> 0\). Before showing the regularity properties of \(\Phi_\varepsilon^j\)’s and their Wick products, we first prove the following useful estimates for \(\widehat\mathcal{D}_\varepsilon(t, n)\) defined in 26 .

We are now ready to show that \(\Phi_\varepsilon^j\) and its Wick products and powers share the same regularity properties with those of \(\Psi_\varepsilon^j\).

6.3 Convergence of invariant Gibbs dynamics↩︎

We now consider the Gibbs dynamics for \(\text{HLSM}_{\varepsilon, N}\): \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^{N, j} = - \frac{1}{N} \sum_{k = 1}^N :\!{ (u_\varepsilon^{N, k})^2 u_\varepsilon^{N, j} }\!: + \, \sqrt{2} \xi^j, \qquad j = 1, \dots, N , \label{HLSMu} \end{align}\tag{147}\]

and the mean-field SNLH: \[\begin{align} (\partial_t+ 1 - \Delta) u^j = - \mathbb{E}[ (u^j)^2 - (\Phi_0^j)^2 ] u^j + \sqrt{2} \xi^j , \label{mfSNLHu} \end{align}\tag{148}\]

where \(\Phi_0^j\) is as defined in 135 . We also write out the two intermediate equations in the two convergence regimes, namely \(\text{PLSM}_N\): \[\begin{align} (\partial_t+ 1 - \Delta) u^{N, j} = - \frac{1}{N} \sum_{k = 1}^N :\!{ (u^{N, k})^2 u^{N, j} }\!: + \, \sqrt{2} \xi^j, \qquad j = 1, \dots, N , \label{PLSMu} \end{align}\tag{149}\]

and the mean-field \(\text{SdNLW}_\varepsilon\): \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) u_\varepsilon^j = - \mathbb{E}[ (u_\varepsilon^j)^2 - (\Phi_\varepsilon^j)^2 ] u_\varepsilon^j + \sqrt{2} \xi^j , \label{mfSdNLWu} \end{align}\tag{150}\]

where \(\Phi_\varepsilon^j\) is as defined in 134 . In the above four equations, we have already renormalized the nonlinearities, and as in the previous sections, we define the solutions to the above four equations as follows. By using the first order expansions: \[\begin{align} \begin{aligned} u_\varepsilon^{N, j} &= \Phi_\varepsilon^j + v_\varepsilon^{N, j}, \qquad j = 1, \dots, N , \\ u^j &= \Phi_0^j + v^j, \qquad j \in \mathbb{N}, \\ u^{N, j} &= \Phi_0^j + v^{N, j}, \qquad j = 1, \dots, N , \\ u_\varepsilon^j &= \Phi_\varepsilon^j + v_\varepsilon^j, \qquad j \in \mathbb{N}, \end{aligned} \end{align}\]

we say that \(\mathbf{u}_\varepsilon^N = (u_\varepsilon^{N, j})_{1 \leq j \leq N}\), \(u^j\), \(\mathbf{u}^N = (u^{N, j})_{1 \leq j \leq N}\), or \(u_\varepsilon^j\) is a solution to \(\mathrm{HLSM}_{\varepsilon, N}\) 147 , the mean-field SNLH 148 , \(\mathrm{PLSM}_N\) 149 , or the mean-field \(\mathrm{SdNLW}_\varepsilon\) 150 , respectively, if \(\mathbf{v}_\varepsilon^N = (v_\varepsilon^{N, j})_{1 \leq j \leq N}\), \(v^j\), \(\mathbf{v}^N = (v^{N, j})_{1 \leq j \leq N}\), or \(v_\varepsilon^j\) satisfies the perturbed \(\text{HLSM}_{\varepsilon, N}\) \[\begin{align} \begin{aligned} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) v_\varepsilon^{N, j} &= - \frac{1}{N} \sum_{k = 1}^N \big( :\!{ (\Phi_\varepsilon^k)^2 \Phi_\varepsilon^j }\!: + :\!{ (\Phi_\varepsilon^k)^2 }\!: v_\varepsilon^{N, j} + 2 v_\varepsilon^{N, k} :\!{ \Phi_\varepsilon^k \Phi_\varepsilon^j }\!: \\ &\,\,\,\, + 2 \Phi_\varepsilon^k v_\varepsilon^{N, k} v_\varepsilon^{N, j} + (v_\varepsilon^{N, k})^2 \Phi_\varepsilon^j + (v_\varepsilon^{N, k})^2 v_\varepsilon^{N, j} \big) , \qquad j = 1, \dots, N , \end{aligned} \label{HLSMv} \end{align}\tag{151}\]

the perturbed mean-field SNLH \[\begin{align} (\partial_t+ 1 - \Delta) v^j = - 2 \mathbb{E}[\Phi_0^j v^j] \Phi_0^j - 2 \mathbb{E}[\Phi_0^j v^j] v^j - \mathbb{E}[(v^j)^2] \Phi_0^j - \mathbb{E}[(v^j)^2] v^j , \label{mfSNLHv} \end{align}\tag{152}\]

the perturbed \(\text{PLSM}_N\) \[\begin{align} \begin{aligned} (\partial_t+ 1 - \Delta) v^{N, j} = - \frac{1}{N} &\sum_{k = 1}^N \big( :\!{ (\Phi_0^k)^2 \Phi_0^j }\!: + :\!{ (\Phi_0^k)^2 }\!: v^{N, j} + 2 v^{N, k} :\!{ \Phi_0^k \Phi_0^j }\!: \\ &\, + 2 \Phi_0^k v^{N, k} v^{N, j} + (v^{N, k})^2 \Phi_0^j + (v^{N, k})^2 v^{N, j} \big) , \qquad j = 1, \dots, N , \end{aligned} \label{PLSMv} \end{align}\tag{153}\]

or the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) \[\begin{align} (\varepsilon^2 \partial_t^2 + \partial_t+ 1 - \Delta) v_\varepsilon^j = - 2 \mathbb{E}[\Phi_\varepsilon^j v_\varepsilon^j] \Phi_\varepsilon^j - 2 \mathbb{E}[\Phi_\varepsilon^j v_\varepsilon^j] v_\varepsilon^j - \mathbb{E}[(v_\varepsilon^j)^2] \Phi_\varepsilon^j - \mathbb{E}[(v_\varepsilon^j)^2] v_\varepsilon^j , \label{mfSdNLWv} \end{align}\tag{154}\]

respectively, where the Wick products and powers of \(\Phi_\varepsilon^j\)’s are defined in 136 .

From Lemma 17, we know that \(\Phi_\varepsilon^j\)’s and their Wick products and powers share the same regularity properties with those from the stochastic convolutions \(\Psi_\varepsilon^j\)’s given by 5 and 13 . Consequently, with \(\Psi_\varepsilon^j\)’s replaced by \(\Phi_\varepsilon^j\)’s, all the global well-posedness results stated in Subsection 1.2 apply to the above four equations: Proposition 1 applies to the renormalized \(\mathrm{HLSM}_{\varepsilon, N}\) 147 , Proposition 2 applies to the renormalized mean-field SNLH 148 , Theorem 3 (i) applies to the renormalized \(\mathrm{PLSM}_N\) 149 , and Theorem 5 (i) applies to the renormalized mean-field \(\mathrm{SdNLW}_\varepsilon\) 150 .

Let us also mention the following invariance results for the above four equations. For each \(0 < \varepsilon\leq 1\), the Gibbs measure \(\vec{\rho}_\varepsilon^N\) defined in 132 is invariant under the dynamics of the renormalized \(\mathrm{HLSM}_{\varepsilon, N}\) 147 in the sense that for each \(t \in \mathbb{R}_+\), we have \[\begin{align} \mathop{\mathrm{Law}}\big( \mathbf{u}_\varepsilon^N (t), \partial_t\mathbf{u}_\varepsilon^N (t) \big) = \vec{\rho}_\varepsilon^N . \end{align}\]

This follows directly from the invariance of the frequency-truncated Gibbs measure \(\vec{\rho}_\varepsilon^N\) under the flow of the frequency-truncated \(\mathrm{HLSM}_{\varepsilon, N}\) 147 (see, for example, [15], [48]) and a PDE approximation argument. In particular, Bourgain’s invariant measure argument in [42], [62] is not needed. Also, the Gibbs measure \(\rho^N\) defined in 131 is invariant under the dynamics of the renormalized \(\mathrm{PLSM}_N\) 149 in the sense that for each \(t \in \mathbb{R}_+\), we have \[\begin{align} \mathop{\mathrm{Law}}\big( \mathbf{u}^N (t) \big) = \rho^N . \end{align}\]

This follows from [4], where we note that the system 149 is the stochastic quantization of the Gibbs measure \(\rho^N\) in 131 . Moreover, in view of Lemma 15, we see that \(u^j = \Phi_0^j\) is a solution to the mean-field SNLH 148 that preserves the Gaussian free field \(\mu_1\) and that, for each \(0 < \varepsilon\leq 1\), \((u_\varepsilon^j, \partial_tu_\varepsilon^j) = (\Phi_\varepsilon^j, \partial_t\Phi_\varepsilon^j)\) is a solution to the mean-field \(\mathrm{SdNLW}_\varepsilon\) 150 that preserves the Gaussian measure \(\mu_1 \otimes \mu_{0, \varepsilon}\).²

In order to discuss convergence of dynamics at Gibbs equilibrium, we need to prepare Gibbsian initial data corresponding to the Gibbs measure \(\rho^N\). The construction of such Gibbsian initial data is provided by the following proposition, which was proved in [1] based on the study of \(\rho^N\) in [63].

Here, we slightly abused notations, since \(\phi_0^j\) and \(\phi_1^j\) are already used as initial data for the stochastic convolutions \(\Phi_\varepsilon^j\) defined in 134 and \(\Phi_0^j\) in 135 . However, with the new initial data \(\{\phi_0^j, \phi_1^j\}_{j \in \mathbb{N}}\) provided in Proposition 16, the statistical properties of the stochastic convolutions \(\Phi_\varepsilon^j\) and \(\Phi_0^j\) remain unchanged. In particular, the family of initial data \(\{\phi_0^j, \phi_1^j\}_{j \in \mathbb{N}}\) provided in Proposition 16 is again assumed to be independent of the family of space-time white noises \(\{\xi^j\}_{j \in \mathbb{N}}\), and so Lemma 15 and Lemma 17 remain true with the new data.

In view of Proposition 16, we can equip the renormalized \(\mathrm{HLSM}_{\varepsilon, N}\) 147 with initial data \(((\phi_0^j + v_0^{N, j}, \varepsilon^{-1} \phi_1^j))_{1 \leq j \leq N}\) and the renormalized \(\mathrm{PLSM}_N\) 149 with initial data \((\phi_0^j + v_0^{N, j})_{1 \leq j \leq N}\). Then, since each \(\Phi_\varepsilon^j\) in 134 has Gaussian initial data \((\phi_0^j, \phi_1^j)\), we see that the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 151 should be equipped with initial data \((\mathbf{v}_0^N, \mathbf{0}) = ((v_0^{N, j}, 0))_{1 \leq j \leq N}\). Similarly, given that each \(\Phi_0^j\) in 135 has Gaussian initial data \(\phi_0^j\), we equip the perturbed \(\mathrm{PLSM}_N\) 153 with initial data \(\mathbf{v}_0^N = (v_0^{N, j})_{1 \leq j \leq N}\). In view of the decay of \(v_0^{N, j}\) in ?? as \(N \to \infty\), we see that the solution \(\mathbf{v}_\varepsilon^N = (v_\varepsilon^{N, j})_{1 \leq j \leq N}\) to the perturbed \(\mathrm{HLSM}_{\varepsilon, N}\) 151 and the solution \(\mathbf{v}^N = (v^{N, j})_{1 \leq j \leq N}\) to the perturbed \(\mathrm{PLSM}_N\) 153 will converge to zero as \(N \to \infty\). This is coherent with the mean-field convergence of these two systems, since \(v^j \equiv 0\) is the unique solution to the perturbed mean-field SNLH 152 and \(v^j_\varepsilon\equiv 0\) is the unique solution to the perturbed mean-field \(\mathrm{SdNLW}_\varepsilon\) 154 , both with zero initial data.

We are now ready to state the convergence results in Figure 1 for invariant Gibbs dynamics. In view of the initial data set from Proposition 16 and the regularity properties of the stochastic objects in Lemma 17, the proof follows from essentially the same steps for the convergence of general dynamics in Theorem 3 (ii) and (iii) and Theorem 5 (ii) and (iii).

References↩︎

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1. The convergence result in [4] is based on the assumptions in [4], which involve an \(L^2 (\Omega)\)-convergence of the initial data. However, we note that in establishing convergence of the solutions in probability, the authors did not use the assumption on the \(L^2 (\Omega)\)-convergence of the initial data.↩︎

2. A direct but tedious computation similar to that in Lemma 15 shows that \(\mathbb{E}[ |\widehat{\partial_t\Phi_\varepsilon^j} (t, n)|^2 ] = \varepsilon^{-2}\) for each \(n \in \mathbb{Z}^2\) and \(t \geq 0\), which shows that \(\mathop{\mathrm{Law}}(\partial_t\Phi_\varepsilon^j (t)) = \mu_{0, \varepsilon}\). Note that each \(\widehat{\partial_t\Phi_\varepsilon^j} (t, n)\) diverges as \(\varepsilon\to 0\), which is compatible with the fact that \(\partial_t\Phi_0^j = - (1 - \Delta) \Phi_0^j + \sqrt{2} \xi^j\) is almost surely merely a distribution in time.↩︎