Laurent Thomann
email: laurent.thomann@univ-lorraine.fr

INVARIANT GIBBS MEASURES FOR THE 2-d DEFOCUSING NONLINEAR WAVE EQUATIONS

Keywords: 2010 Mathematics Subject Classification. 35L71 nonlinear wave equation, nonlinear Klein-Gordon equation, Gibbs measure, Wick ordering, Hermite polynomial, white noise functional, weak universality

We consider the defocusing nonlinear wave equations (NLW) on the twodimensional torus. In particular, we construct invariant Gibbs measures for the renormalized so-called Wick ordered NLW. We then prove weak universality of the Wick ordered NLW, showing that the Wick ordered NLW naturally appears as a suitable scaling limit of non-renormalized NLW with Gaussian random initial data.

Résumé. On considère les équations des ondes non-linéaires défocalisantes sur le tore de dimension deux. On construit des mesures de Gibbs invariantes pour les équations renormalisées au sens de Wick. On prouve ensuite une propriété d'universalité faible pour ces équations renormalisées, en montrant qu'elle apparaissent comme limites d'équations d'ondes non renormalisées avec conditions initiales aléatoires de loi gaussienne.

1. Introduction 1.1. Nonlinear wave equations. We consider the defocusing nonlinear wave equations (NLW) in two spatial dimensions:

∂ 2 t u -∆u + ρu + u 2m+1 = 0 (u, ∂ t u)| t=0 = (φ 0 , φ 1 ), (t, x) ∈ R × M, (1.1)
where ρ ≥ 0 and m ∈ N. When ρ > 0, (1.1) is also referred to as the nonlinear Klein-Gordon equation. We, however, simply refer to (1.1) as NLW and moreover restrict our attention to the real-valued setting. In the following, we mainly consider (1.1) on the twodimensional torus M = T 2 = (R/Z) 2 but we also provide a brief discussion when M is a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R 2 (with the Dirichlet or Neumann boundary condition). See Theorem 1.7 below.

Our main goal in this paper is to construct an invariant Gibbs measure for a renormalized version of (1.1) by studying dynamical properties of the renormalized equation.

∂ t u v = 0 1 -1 0 ∂H ∂(u, v)
,

where H = H(u, v) is the Hamiltonian given by

H(u, v) = 1 2 ˆT2 ρu 2 + |∇u| 2 dx + 1 2 ˆT2 v 2 dx + 1 2m + 2 ˆT2 u 2m+2 dx. (1.2)
By drawing an analogy to the finite dimensional setting, the Hamiltonian structure of the equation and the conservation of the Hamiltonian suggest that the Gibbs measure P (2m+2) 2

of the form: 1 "dP

(2m+2) 2 = Z -1 exp(-βH(u, v))du ⊗ dv" (1.3)
is invariant under the dynamics of (1.1). 2 With (1.2), we can rewrite the formal expression (1.3) as

dP (2m+2) 2 = Z -1 e -1 2m+2
´u2m+2 dx e -1 2 ´(ρu 2 +|∇u| 2 )dx du ⊗ e -1 2 ´v2 dx dv

∼ e -1 2m+2
´u2m+2 dx dµ, (1.4) where µ is the Gaussian measure µ on D ′ (T 2 ) × D ′ (T 2 ) with the density

3 dµ = Z -1 e -1 2
´(ρu 2 +|∇u| 2 )dx du ⊗ e -1 2 ´v2 dx dv.

(1.5)

Note that µ has a tensorial structure: µ = µ 0 ⊗ µ 1 , where the marginal measures µ 0 and µ 1 are given by

dµ 0 = Z -1 0 e -1 2
´(ρu 2 +|∇u| 2 )dx du and

dµ 1 = Z -1 1 e -1 2
´v2 dx dv.

(1.6)

Namely, µ 0 is the Ornstein-Uhlenbeck measure and µ 1 is the white noise measure on T 2 . Recall that µ is the induced probability measure under the map:

4 ω ∈ Ω -→ (u, v) = n∈Z 2 g 0,n (ω) n ρ e in•x , n∈Z 2 g 1,n (ω)e in•x , (1.7)
where n ρ = ρ + |n| 2 and {g 0,n , g 1,n } n∈Z 2 is a sequence of independent standard complex-valued Gaussian random variables on a probability space (Ω, F, P ) conditioned that g j,-n = g j,n , n ∈ Z 2 , j = 0, 1. In view of (1.7), it is easy to see that µ is supported on H s (T 2 ) := H s (T 2 ) × H s-1 (T 2 ), s < 0.

Moreover, we have µ(H 0 (T 2 )) = 0. This implies that ´u2m+2 dx = ∞ almost surely with respect to µ. In particular, the right-hand side of (1.4) would not be a probability measure, thus requiring a renormalization of the potential part of the Hamiltonian. In the twodimensional case, it is known that a Wick ordering suffices for this purpose. See Simon [START_REF] Simon | The P (ϕ)2 Euclidean (quantum) field theory[END_REF] and Glimm-Jaffe [START_REF] Glimm | Quantum physics. A functional integral point of view[END_REF]. Also, see Da Prato-Tubaro [START_REF] Da Prato | Wick powers in stochastic PDEs: an introduction[END_REF] for a concise discussion on T 2 , where the Gibbs measures naturally appear in the context of the stochastic quantization equation.

In the following, we give a brief review of the Wick renormalization on T 2 . See [START_REF] Da Prato | Wick powers in stochastic PDEs: an introduction[END_REF] for more details. Let u denote a typical element under µ 0 defined in (1.6). Since u / ∈ L 2 (T 2 ) almost surely, we have ˆT2 u 2 dx = lim N →∞ ˆT2 (P N u) 2 dx = ∞ almost surely, where P N is the Dirichlet projection onto the frequencies {|n| ≤ N }.

1. Henceforth, we use Z, ZN , etc. to denote various normalizing constants so that the corresponding measures are probability measures when appropriate.

2. We simply set β = 1 in the following. While our analysis holds for any β > 0, the resulting (renormalized) Gibbs measures are mutually singular for different values of β > 0. See [START_REF] Oh | On Cameron-Martin theorem and almost sure global existence[END_REF].

3. On T 2 , we need to assume ρ > 0 in order to avoid a problem at the zeroth frequency. See (1.7) below. In the case of a bounded domain in R 2 with the Dirichlet boundary condition, we can take ρ = 0.

4. We drop the harmless factor 2π in the following.

For each x ∈ T 2 , P N u(x) is a mean-zero real-valued Gaussian random variable with variance 5

σ N def = E[(P N u) 2 (x)] = |n|≤N 1 ρ + |n| 2 ∼ log N. (1.8)
This motivates us to define the Wick ordered monomial : (P N u) k : by

: (P N u) k (x) : = H k (P N u(x); σ N ) (1.9)
in a pointwise manner. Here, H k (x; σ) is the Hermite polynomial of degree k defined in (2.1). Then, with (1.7) and (1.8), it is easy to see that the random variables X N (u) defined by

X N (u) = ˆT2 : (P N u) 2 (x) : dx
have uniformly bounded second moments and converge to some random variable in L 2 (dµ 0 ) which we denote by

X ∞ (u) = ˆT2 : u 2 : dx ∈ L 2 (dµ 0 ).
In view of the Wiener chaos estimate (Lemma 2.2), we see that X N (u) also converges to

X ∞ (u) in L p (dµ 0 ), p < ∞.
In general, given any m ∈ N, one can show that the limit

ˆT2 : u 2m+2 : dx = lim N →∞ ˆT2 : (P N u) 2m+2 : dx (1.10)
exists in L p (µ) for any finite p ≥ 1. Moreover, we have the following proposition.

Proposition 1.1. Let m ∈ N. Then, R N (u) def = e -1 2m+2 ´T2 :(P N u) 2m+2 : dx ∈ L p (µ) for any finite p ≥ 1 with a uniform bound in N , depending on p ≥ 1. Moreover, for any finite

p ≥ 1, R N (u) converges to some R(u) in L p (µ) as N → ∞.
This proposition follows from the hypercontractivity of the Ornstein-Uhlenbeck semigroup and Nelson's estimate [START_REF] Nelson | A quartic interaction in two dimensions[END_REF]. See also [START_REF] Da Prato | Wick powers in stochastic PDEs: an introduction[END_REF][START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF]. Denoting the limit R(u) ∈ L p (µ) by

R(u) = e -
H Wick (u, v) = 1 2 ˆT2 ρu 2 + |∇u| 2 dx + 1 2 ˆT2 v 2 dx + 1 2m + 2 ˆT2 : u 2m+2 : dx by dP (2m+2) 2 = Z -1 e -H Wick (u,v) du ⊗ dv = Z -1 e -1 2m+2 ´T2 :u 2m+2 : dx dµ = Z -1 R(u)dµ.
5. Note that σN defined in (1.8) is independent of x ∈ T 2 . When M is a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R 2 , the variance σN (x) = E[(PN u) 2 (x)] depends on x ∈ M but satisfies the logarithmic bound in N . See (1.12) below.

It follows from Proposition 1.1 that P (2m+2) 2 ≪ µ and, in particular, P (2m+2) 2 is a probability measure on H s (T 2 ) \ H 0 (T 2 ), s < 0. Moreover, defining

P (2m+2) 2,N by dP (2m+2) 2,N = Z -1 N R N (u)dµ,
we see that

P (2m+2) 2,N
converges "uniformly" to

P (2m+2) 2
in the sense that given any ε > 0, there exists N 0 ∈ N such that

P (2m+2) 2,N (A) -P (2m+2) 2 (A) < ε
for any N ≥ N 0 and any measurable set A ⊂ H s (T), s < 0.

Lastly, let us briefly discuss the construction of the Gibbs measure

P (2m+2) 2
when M is a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R 2 (with the Dirichlet or Neumann boundary condition). In this case, the Gaussian measure µ in (1.5) represents the induced probability measure under the map:

ω ∈ Ω -→ (u, v) = n∈N g 0,n (ω) (ρ + λ 2 n ) 1 2 ϕ n (x), n∈N g 1,n (ω)ϕ n (x) , (1.11)
where {ϕ n } n∈N is an orthonormal basis of L 2 (M) consisting of eigenfunctions of the Laplace-Beltrami operator -∆ with the corresponding eigenvalues {λ 2 n } n∈N , which we assume to be arranged in the increasing order. It is easy to see from (1.11) that µ is supported on H s (M) \ H 0 (M), s < 0.

Given N ∈ N, we define σ N by

σ N (x) = E[(P N u N ) 2 (x)] = λn≤N ϕ n (x) 2 ρ + λ 2 n log N, (1.12)
where P N denotes the spectral projector defined by

P N u = λn≤N u(n)ϕ n .
Note that unlike the situation on T 2 , σ N (x) now depends on x ∈ M. The last inequality in (1.12), however, holds independently of x ∈ M thanks to Weyl's law λ n ≈ n 1 2 (see [START_REF] Zworski | Semiclassical analysis[END_REF]Chapter 14]) and [START_REF] Burq | Remarks on the Gibbs measures for nonlinear dispersive equations[END_REF]Proposition 8.1]. With this definition of σ N (x), we can define the Wick ordered monomials : (P N u) k : as in (1.9) and : u k : by the limiting procedure. Then, the discussion above for T 2 , in particular Proposition 1.1, also holds on M. See Section 4 of [START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF]. While the presentation in [START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF] is given in the complex-valued setting, a straightforward modification yields the corresponding result for the real-valued setting.

In the next subsection, we discuss the dynamical problem. Our main goal in this paper is to construct dynamics for the renormalized equation associated with the Wick ordered Hamiltonian H Wick with initial data distributed according to the Gibbs measure P (2m+2) 2 . 1.3. Dynamical problem: Wick ordered NLW. We now consider the following dynamical problem on T 2 associated with the Wick ordered Hamiltonian:

       ∂ t u v = 0 1 -1 0 ∂H Wick ∂(u, v) (u, v)| t=0 = (φ ω 0 , φ ω 1 ), (1.13)
where the initial data (φ ω 0 , φ ω 1 ) is distributed according to the Gibbs measure P (2m+2) 2

. In view of the absolute continuity of P (2m+2) 2 with respect to the Gaussian measure µ (Proposition 1.1), we consider the random initial data (φ ω 0 , φ ω 1 ) distributed according to µ in the following discussion. Namely, we assume that

φ ω 0 , φ ω 1 = n∈Z 2 g 0,n (ω) n ρ e in•x , n∈Z 2 g 1,n (ω)e in•x , (1.14)
where {g 0,n , g 1,n } n∈Z 2 is as in (1.7). Note that, at this point, the potential part 1 2m+2 ´T2 : u 2m+2 : dx of the Wick ordered Hamiltonian is defined only for u distributed according to the Gaussian measure µ via (1.10). In the following, we extend this definition to a wider class of functions in order to treat the Cauchy problem (1.13).

Given N ∈ N, define the truncated Wick ordered Hamiltonian H N Wick by

H N Wick (u, v) = 1 2 ˆT2 ρu 2 + |∇u| 2 dx + 1 2 ˆT2 v 2 dx + 1 2m + 2 ˆT2 : (P N u) 2m+2 : dx (1.15)
and consider the associated Hamiltonian dynamics:

       ∂ t u N v N = 0 1 -1 0 ∂H N Wick ∂(u N , v N ) (u N , v N )| t=0 = (φ ω 0 , φ ω 1
). Thanks to (1.9) and ∂ x H k (x; σ) = kH k-1 (x; σ), we can rewrite the system (1.15) as the following truncated Wick ordered NLW:

∂ 2 t u N -∆u N + ρu N + P N : (P N u N ) 2m+1 : = 0 (u N , ∂ t u N )| t=0 = (φ ω 0 , φ ω 1 ), (1.16)
where the truncated Wick ordered nonlinearity is interpreted as

P N : (P N u N ) 2m+1 : = P N H 2m+1 (P N u N ; σ N ) .
Let z = z ω denote the random linear solution:

z(t) = S(t)(φ ω 0 , φ ω 1 ) = cos(t ∇ ρ )φ ω 0 + sin(t ∇ ρ ) ∇ ρ φ ω 1 , (1.17)
where ∇ ρ = √ ρ -∆. In view of the Duhamel formula, it is natural to decompose the solution u N to (1.16) as

u N = z + w N .
Note that we have P N w N = w N . By recalling the following identities for the Hermite polynomials:

H k (x + y) = k ℓ=0 k ℓ H ℓ (y) • x k-ℓ and H k (x; σ) = σ k 2 H k (σ -1 2 x), (1.18)
we have

: (P N u N ) 2m+1 : = H 2m+1 (z N + w N ; σ N ) = 2m+1 ℓ=0 2m + 1 ℓ H ℓ (z N ; σ N ) • w 2m+1-ℓ N , (1.19)
where z N = P N z. This shows that applying the Wick ordering to the monomial

(P N u N ) 2m+1 = (z N + w N ) 2m+1 = 2m+1 ℓ=0 2m + 1 ℓ z ℓ N • w 2m+1-ℓ N (1.20)
is equivalent to Wick ordering all the monomials z ℓ N . Namely, replacing each z ℓ in (1.20) by : z ℓ N : = H ℓ (z N ; σ N ) yields the Wick ordered monomial : (P N u N ) 2m+1 : via (1.19). In Proposition 2.3 below, we prove that :

z ℓ N : ∈ L p (Ω; L q ([-T, T ]; W -ε,r (T 2 )
)) for any p, q, r < ∞, T > 0, and ε > 0 with a bound uniform in N . Moreover, the sequence : z ℓ N : N ∈N is a Cauchy sequence in the same space, thus allowing us to define

: z ℓ : = : z ℓ ∞ : def = lim N →∞ : z ℓ N : (1.21) in L p (Ω; L q ([-T, T ]; W -ε,r (T 2 )
)) for any p, q, r < ∞, T > 0, and ε > 0 (and for any ℓ ∈ N). Now, consider a function u of the form

u = z + w (1.22)
for some "nice" w. Then, we can use (1.18) and (1.21) to define the Wick ordered monomial : u 2m+1 : for functions u of the form (1.22) by : u 2m+1 : = : (z + w) 2m+1 : =

2m+1 ℓ=0 2m + 1 ℓ : z ℓ : • w 2m+1-ℓ . (1.23)
Hence, we finally arrive at the defocusing Wick ordered NLW:

∂ 2 t u -∆u + ρu + : u 2m+1 : = 0 (u, ∂ t u)| t=0 = (φ ω 0 , φ ω 1 ), (1.24)
where (φ ω 0 , φ ω 1 ) is as in (1.14). Before we state our main result, we first recall two critical regularities associated with (1.1) on R 2 with ρ = 0. On the one hand, the scaling symmetry for (1.1) induces the so-called scaling critical Sobolev index:

s 1 = 1 -1 m .
On the other hand, the Lorentzian invariance (conformal symmetry) induces another critical regularity: s 2 = 3 4 -1 2m (at least in the focusing case). Hence, we set s crit by

s crit = max 1 - 1 m , 3 4 - 1 2m = 1 4 if m = 1, 1 -1 m if m ≥ 2.
We now state our main result.

Theorem 1.2. Let M = T 2 , m ∈ N, and ρ > 0. Then, the Wick ordered NLW (1.24) is almost surely locally well-posed with respect to the Gaussian measure µ defined in (1.5). More precisely, letting (φ ω 0 , φ ω 1 ) be as in (1.14), there exist C, c > 0 such that for each T ≪ 1, there exists a set Ω T ⊂ Ω with the following properties:

(i) P (Ω c T ) ≤ C exp -1 T c , (ii) For each ω ∈ Ω T , there exists a (unique) solution u to (1.24) with (u, ∂ t u)| t=0 = (φ ω 0 , φ ω 1 ) in the class S(t)(φ ω 0 , φ ω 1 ) + C([-T, T ]; H s (T 2 )) ∩ X s, 1 2 + T ⊂ C([-T, T ]; H -ε (T 2 ))
for any s ∈ (s crit , 1) and ε > 0. Here, X s, 1 2 + T denotes the local-in-time version of the hyperbolic Sobolev space. See Section 3.

We emphasize that the Wick ordered NLW (1.24) is defined only for functions u of the form (1.22). Then, the residual term w = u -z satisfies the following perturbed Wick ordered NLW:

∂ 2 t w -∆w + ρw + : (w + z) 2m+1 : = 0 (w, ∂ t w)| t=0 = (0, 0). (1.25)
By writing (1.25) in the Duhamel formulation, we obtain

w(t) = - ˆt 0 sin((t -t ′ ) ∇ ρ ) ∇ ρ : (w + z) 2m+1 (t ′ ) : dt ′ = - 2m+1 ℓ=0 ˆt 0 sin((t -t ′ ) ∇ ρ ) ∇ ρ 2m + 1 ℓ : z ℓ (t ′ ) : • w 2m+1-ℓ (t ′ )dt ′ . (1.26)
We prove Theorem 1.2 by solving the fixed point problem (1.26) for w in

C([-T, T ]; H s (T 2 )) ∩ X s, 1 2 + T , s > s crit .
In Section 2, we study the regularity of the random linear solution z and the associated Wick ordered monomials : z ℓ :. In particular, while they are rough, : z ℓ : enjoys enhanced integrability both in space and time. See Proposition 2.3. In Section 3, we then use the standard Fourier restriction norm method to solve the fixed point problem (1.26). The original idea of this argument with the decomposition (1.22) appears in McKean [START_REF] Mckean | Statistical mechanics of nonlinear wave equations[END_REF] and Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] in the context of the nonlinear Schrödinger equations on T d , d = 1, 2. See also Burq-Tzvetkov [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. I. Local theory[END_REF]. In the field of the stochastic PDEs, this method is known as Da Prato-Debussche trick [START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a space-time white noise[END_REF].

Remark 1.3. As in the study of singular stochastic PDEs, our proof consists of factorizing the ill-defined solution map: (φ ω 0 , φ ω 1 ) → u into a canonical lift followed by a (continuous) solutions map Ψ:

(φ ω 0 , φ ω 1 ) lift -→ (z ω 1 , z ω 3 , . . . , z ω 2m+1 ) Ψ -→ w ∈ C([-T, T ]; H s (T 2 )) -→ u = z + w ∈ C([-T, T ]; H -ε (T 2 )),
for s ∈ (s crit , 1) and ε > 0, where z k def = : z k : . On the one hand, we use probability theory to construct the data set {z 2j+1 } m j=0 in the first step. On the other hand, the second step is entirely deterministic. Moreover, the solution map Ψ in the second step is continuous

from m j=0 S j T to X s, 1 2 + T
, where S j T denotes some appropriate Strichartz space for z 2j+1 . See Section 3.

Remark 1.4. The same almost sure local well-posedness holds for the truncated Wick ordered NLW (1.16). More precisely, we can choose Ω T , independent of N ∈ N, such that the statement in Theorem 1.2 holds for (1.24) and (1.16). Moreover, by possibly shrinking the time, one can also prove that the solution u N = u ω N to (1.16) converges to the solution u = u ω to (1.24) as N → ∞.

Once we have almost sure local well-posedness of (1.24), the invariant measure argument by Bourgain [START_REF] Bourgain | Periodic nonlinear Schrödinger equation and invariant measures[END_REF][START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] yields the following almost sure global well-posedness of (1.24) and invariance of the Gibbs measure P is invariant under the dynamics of (1.24).

The proof of Theorem 1.5 exploits the invariance of the truncated Gibbs measure

P (2m+2) 2,N
for the truncated Wick ordered NLW (1.16) and combines it with an approximation argument. See Remark 1.4. As this argument is standard by now, we omit the proof. See Bourgain [START_REF] Bourgain | Invariant measures for the 2D-defocusing nonlinear Schrödinger equation[END_REF] and Burq-Tzvetkov [START_REF] Burq | Random data Cauchy theory for supercritical wave equations. II. A global existence result[END_REF] for details.

Remark 1.6. We point that the convergence result in Remark 1.4 and invariance of the Gibbs measure in Theorem 1.5 already appear (without a proof) in the lecture note by Bourgain [START_REF] Bourgain | Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions[END_REF]. See [5, Theorem 111 on p. 63] and a comment that follows (118) on p. 64 in [START_REF] Bourgain | Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions[END_REF]. To the best of our knowledge, however, there seems to be no proof available in a published paper. In fact, one of the main purposes of this paper is to present the details of the proof of Bourgain's claim in [START_REF] Bourgain | Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions[END_REF].

Next, we briefly discuss the situation when the spatial domain M is a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R 2 (with the Dirichlet or Neumann boundary condition). In this case, one can exploit the invariance of the truncated Gibbs measures P has a global-in-time solution u ∈ C(R; H s (M)) for any s < 0. Moreover, for all t ∈ R, the law of the random function (u, ∂ t u)(t) is given by P (2m+2) 2

.

In [START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF], we proved an analogous result for the defocusing Wick ordered nonlinear Schrödinger equations on M. Theorem 1.7 follows from repeating the argument presented in [START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF] with systematic modifications and thus we omit details. See also [START_REF] Albeverio | Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids[END_REF][START_REF] Da Prato | Two-dimensional Navier-Stokes equations driven by a space-time white noise[END_REF][START_REF] Burq | Remarks on the Gibbs measures for nonlinear dispersive equations[END_REF][START_REF] Oh | On invariant Gibbs measures for the generalized KdV equations[END_REF]. The main ingredient for Theorem 1.7 is to establish tightness (= compactness) of measures ν N on space-time functions, emanating from the truncated Gibbs measure P (2m+2) 2,N and then upgrading the weak convergence of ν N (up to a subsequence) to an almost sure convergence of the corresponding random variables via Skorokhod's theorem. Due to the compactness argument, Theorem 1.7 claims only the existence of a global-in-time solution u. Lastly, note that Theorem 1.7 only claims that the law of the H s -valued random variable (u, ∂ t u)(t) is given by the Gibbs measure P (2m+2) 2 for any t ∈ R. In particular, this mild invariance for a general geometric setting is weaker than the invariance stated in Theorem 1.5 for the Wick ordered NLW (1.24) on T 2 . Remark 1.8. On the one hand, the defocusing/focusing nature of the equation does not play any role in the almost sure local well-posedness result (Theorem 1.2) and thus Theorem 1.2 also holds in the focusing case. It can also be extended to Wick ordered even power monomials in the equation. On the other hand, the defocusing nature of the equation plays a crucial role in the proof of Proposition 1.1 and hence in Theorems 1.5 and 1.7. In the focusing case (i.e. with -u 2m+1 , m ∈ N, in (1.1)), it is known that the Gibbs measure can not be normalized in the two dimensional case. See Brydges-Slade [START_REF] Brydges | Statistical mechanics of the 2-dimensional focusing nonlinear Schrödinger equation[END_REF]. Lastly, we point out that in the case of the quadratic nonlinearity (which is neither defocusing nor focusing), one can introduce the following modified Gibbs measure:

dP (3) 2 = Z -1 e -1 3
´:u 3 : -A( ´:u 2 : ) 2 dµ for sufficiently large A ≫ 1 and study the associated dynamical problem. See [START_REF] Bourgain | Nonlinear Schrödinger equations, Hyperbolic equations and frequency interactions[END_REF] for the construction of this modified Gibbs measure P

2 . 1.4. Wick ordered NLW as a scaling limit. As an application of the local wellposedness argument, we show how the Wick ordered NLW (1.24) appears as a scaling limit of non-renormalized NLW equations on dilated tori. This part of the discussion is strongly motivated by the weak universality result for the Wick ordered stochastic NLW on T 2 studied by the first author with Gubinelli and Koch in [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equations[END_REF].

Fix ρ > 0. Given small ε > 0, we consider the following non-renormalized NLW equation on a dilated torus T 2

ε def = (ε -1 T) 2 : ∂ 2 t v ε -∆v ε + ρ ε v ε = f (v ε ) (v ε , ∂ t v ε )| t=0 = (ψ ω ε,0 , ψ ω ε,1 ), (t, x) ∈ R × T 2 ε (1.27)
with Gaussian random initial data (ψ ω ε,0 , ψ ω ε,1 ), where f : R → R is a smooth odd 6 function with the following bound:

|f (4) (x)| 1 + |x| M (1.28)
for some M ≥ 0 and ρ ε is a parameter to be chosen later. In the following, we choose ψ ω ε,0 and ψ ω ε,1 to be a smoothed Ornstein-Uhlenbeck process and a smoothed white noise on T 2 ε , respectively. For the sake of concreteness, we set 7

(ψ ω ε,0 , ψ ω ε,1 ) = n∈(εZ) 2 |n|≤1 g 0,ε -1 n ε -1 ε 2 ρ + |n| 2 e in•x ε -1 , n∈(εZ) 2 |n|≤1 g 1,ε -1 n e in•x ε -1 , 6.
It follows from the proof of Theorem 1.9 that it suffices to assume that f (0) = f ′′ (0) = 0 for the cubic case considered in Theorem 1.9.

7. Note that εe in•x n∈(εZ) 2 forms an orthonormal basis of L 2 (T 2 ε ). Moreover, recall that the Fourier-Wiener series

n∈(εZ) 2 g0,n ε -1 |n| e in•x ε -1
represents the periodic Wiener process on T 2 ε .

where {g 0,n , g 1,n } n∈Z 2 is as in (1.7). Our main goal is to study the behavior of the solution to (1.27) as ε → 0 by applying a suitable scaling.

Let u ε (t, x)

def = ε -1 v ε (ε -1 x, ε -1 t). Then, u ε satisfies ∂ 2 t u ε -∆u ε + ρu ε = ε -3 f (εu ε ) + ε(ε 2 ρ -ρ ε )u ε (u ε , ∂ t u ε )| t=0 = (φ ω ε,0 , φ ω ε,1 ), (t, x) ∈ R × T 2 , (1.29)
where (φ ω ε,0 , φ ω ε,1 ) on T 2 is given by

(φ ω ε,0 , φ ω ε,1 ) = n∈Z 2 |n|≤ε -1 g 0,n ρ + |n| 2 e in•x , n∈Z 2 |n|≤ε -1 g 1,n e in•x .
(1.30)

Note that (φ ω ε,0 , φ ω ε,1 ) converges to (φ ω 0 , φ ω 1 ) in (1.14) distributed according to the Gaussian measure µ defined in (1.5).

The rescaled model (1.29) on T 2 allows us to study the large temporal and spatial scale behavior of the solution v ε to (1.27). Moreover, by suitably choosing ρ ε , the family {u ε } ε>0 converges to the solution u to the Wick ordered NLW on T 2 with a parameter λ = λ(f ), depending only on f . Theorem 1.9. Let ρ > 0. Then, there exists a choice of ρ ε such that, as ε → 0, the family of the solutions {u ε } ε>0 to (1.29) converges almost surely to the solution u to the following Wick ordered cubic NLW on T 2 :

∂ 2 t u -∆u + ρu = λ : u 3 : (u, ∂ t u)| t=0 = (φ ω 0 , φ ω 1 ), (1.31)
where the convergence takes place in C([-T ω , T ω ]; H s (T 2 )), s < 0, for some T ω > 0. Here, the constant λ is given by λ = f (3) (0)

6

, depending only on the function f .

This theorem shows a kind of weak universality for the Wick ordered NLW. See [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equations[END_REF] for a similar result for the Wick ordered stochastic NLW. We also refer readers to [START_REF] Hairer | A class of growth models rescaling to KPZ[END_REF][START_REF] Gubinelli | [END_REF][START_REF] Gubinelli | The Hairer-Quastel universality result at stationarity[END_REF] for more discussion on weak universality (for stochastic parabolic equations, in particular the KPZ equation).

Remark 1.10. By starting with the following NLW on T 2 ε :

∂ 2 t v ε -∆v ε + ρ ε v ε + m-1 j=1 a j (ε)v 2j+1 ε = f (v ε ) (v ε , ∂ t v ε )| t=0 = (ψ ω ε,0 , ψ ω ε,1 ),
we can tune the m parameters ρ ε , a j (ε), j = 1, . . . , m -1, such that by a small modification of the proof of Theorem 1.9, we obtain the following Wick ordered NLW:

∂ 2 t u -∆u + ρu = λ : u 2m+1 : (u, ∂ t u)| t=0 = (φ ω 0 , φ ω 1 ),
for some λ = λ(f ), as ε → 0. In this case, one needs to use the scaling u ε (t, x) = ε -γ v ε (ε -1 x, ε -1 t) for some suitably chosen γ = γ(m) > 0 and also assume a bound analogous to (1.28) for a higher order derivative of f .

Probabilistic tools

In this section, we first recall basic probabilistic tools. Then, we prove a uniform (in N ) bound on the Wick ordered monomials : z ℓ N : = H ℓ (z N , σ N ), consisting of the random linear solution (Proposition 2.3). Moreover, we prove that : z ℓ N : N ∈N is a Cauchy sequence, allowing us to define : z ℓ : by (1.21).

2.1. Hermite polynomials and white noise functional. First, recall the Hermite polynomials H k (x; σ) defined via the generating function:

F (t, x; σ) := e tx-1 2 σt 2 = ∞ k=0 t k k! H k (x; σ). (2.1)
For simplicity, we set H k (x) := H k (x; 1). In the following, we list the first few Hermite polynomials for readers' convenience:

H 0 (x; σ) = 1, H 1 (x; σ) = x, H 2 (x; σ) = x 2 -σ, H 3 (x; σ) = x 3 -3σx, H 4 (x; σ) = x 4 -6σx 2 + 3σ 2 . (2.2)
Next, we define the white noise functional. Let ξ(x; ω) be the (real-valued) mean-zero Gaussian white noise on T 2 defined by

ξ(x; ω) = n∈Z 2 g n (ω)e in•x ,
where {g n } n∈Z 2 is a sequence of independent standard complex-valued Gaussian random variables conditioned that g -n = g n , n ∈ Z 2 . It is easy to see that ξ ∈ H s (T 2 ) \ H -1 (T 2 ), s < -1, almost surely. In particular, ξ is a distribution, acting on smooth functions. In fact, the action of ξ can be defined on L 2 (T 2 ).

We define the white noise functional

W (•) : L 2 (T 2 ) → L 2 (Ω) by W f (ω) = f, ξ(ω) L 2 = n∈Z 2 f (n)g n (ω) (2.3)
for a real-valued function f ∈ L 2 (T 2 ). Note that W f = ξ(f ) is basically the Wiener integral of f . In particular, W f is a real-valued Gaussian random variable with mean 0 and variance

f 2 L 2 . Moreover, W (•) is unitary: E W f W h ] = f, h L 2 x (2.4) for f, h ∈ L 2 (T 2
). The following lemma extends the relation (2.4) to a more general setting.

Lemma 2.1. Let f, h ∈ L 2 (T 2 ) such that f L 2 = h L 2 = 1. Then, for k, m ∈ Z ≥0 , we have E H k (W f )H m (W h ) = δ km k![ f, h ] k .
Here, δ km denotes the Kronecker's delta function.

This lemma follows from computing the left-hand side of

E[F (t, W f )F (s, W h )] = ∞ k,m=0 t k k! s m m! E H k (W f )H m (W h )
and comparing the coefficients. See [START_REF] Da Prato | Wick powers in stochastic PDEs: an introduction[END_REF][START_REF] Oh | On invariant Gibbs measures for the generalized KdV equations[END_REF] for details.

We also recall the following Wiener chaos estimate [START_REF] Simon | The P (ϕ)2 Euclidean (quantum) field theory[END_REF]Theorem I.22].

Lemma 2.2. Fix k ∈ N and c(n 1 , . . . , n k ) ∈ C. Given d ∈ N, let {g n } d n=1 be a sequence of independent standard complex-valued Gaussian random variables and set g -n = g n . Define S k (ω) by

S k (ω) = Γ(k,d) c(n 1 , . . . , n k )g n 1 (ω) • • • g n k (ω),
where Γ(k, d) is defined by

Γ(k, d) = (n 1 , . . . , n k ) ∈ {0, ±1, . . . , ±d} k .
Then, for p ≥ 2, we have

S k L p (Ω) ≤ (p -1) k 2 S k L 2 (Ω) .
(2.5)

The crucial point is that the constant in (2.5) is independent of d ∈ N. This lemma is a direct corollary to the hypercontractivity of the Ornstein-Uhlenbeck semigroup due to Nelson [START_REF] Nelson | A quartic interaction in two dimensions[END_REF].

Stochastic estimate on Wick ordered monomials.

In this subsection, we study the Wick ordered monomials : z ℓ N : and : z ℓ :, consisting of the random linear solution z defined in (1.17). From (1.14) and (1.17), we have

z(t, n) = cos(t n ρ ) n ρ g 0,n + sin(t n ρ ) n ρ g 1,n . (2.6)
In order to avoid the combinatorial complexity in higher ordered monomials, we use the white noise functional as in [START_REF] Oh | A pedestrian approach to the invariant Gibbs measure for the 2-d defocusing nonlinear Schrödinger equations[END_REF]. We, however, need to adapt the white noise functional to z(t). In view of (2.6), we define the white noise functional W t (•) : L 2 (T 2 ) → L 2 (Ω) with a parameter t ∈ R by

W t f (ω) = f, ξ t (ω) L 2 = n∈Z 2 f (n)g t n (ω). (2.7)
Here, ξ t denotes (a specific realization of) the white noise on T 2 given by

ξ t (x; ω) = n∈Z 2 g t n (ω)e in•x ,
where g t n is define by

g t n = cos(t n ρ )g 0,n + sin(t n ρ )g 1,n . Note that E[g t n ] = 0 and Var(g t n ) = cos 2 (t n ρ )+sin 2 (t n ρ ) = 1.
Thus, for each fixed t ∈ R, {g t n } n∈Z 2 is a sequence of independent standard Gaussian random variables conditioned that g t -n = g t n for all n ∈ N. Therefore, the white noise functional W t (•) defined in (2.7) satisfies the same properties as the standard white noise functional W (•) defined in (2.3). Lastly, note that, in view of (2.6), the random linear solution z N = P N z can be expressed as

z N (t, x) = |n|≤N g t n (ω) n ρ e in•x . (2.8)
In the following, we use the short-hand notation L q T = L q t ([-T, T ]), etc.

Proposition 2.3. Let ℓ ∈ N and ρ > 0. Then, given 2 ≤ q, r < ∞ and ε > 0, there exist C, c > 0 such that

P ∇ -ε : z ℓ N : L q T L r x > λ ≤ C exp -c λ 2 ℓ T 2 qℓ
(2.9)

for any T > 0, λ > 0, and any N ∈ N. Moreover, the sequence : z ℓ N : N ∈N is a Cauchy sequence in L p (Ω; L q ([-T, T ]; W -ε,r (T 2 ))). In particular, denoting the limit by : z ℓ :, we have : z ℓ : ∈ L q ([-T, T ]; W -ε,r (T 2 )) almost surely, satisfying the tail estimate (2.9).

Before proceeding to the proof of Proposition 2.3, we introduce some notations. Let σ N be as in (1.8). For fixed x ∈ T 2 and N ∈ N, we also define

η N (x)(•) def = 1 σ 1 2 N |n|≤N e n (x) n ρ e n (•) and γ N (•) def = |n|≤N 1 n 2 ρ e n (•), (2.10)
where e n (y) = e in•y . Note that η N (x)(•) is real-valued with η N (x) L 2 (T 2 ) = 1 for all x ∈ T 2 and all N ∈ N. Moreover, we have

η M (x), η N (y) L 2 = 1 σ 1 2 M σ 1 2 N γ N (y -x) = 1 σ 1 2 M σ 1 2 N γ N (x -y), (2.11)
for fixed x, y ∈ T 2 and M ≥ N ≥ 1.

Proof. From (2.8) and (2.10), we have

z N (t, x) = σ 1 2 N z N (t, x) σ 1 2 N = σ 1 2 N W t η N (x) = σ 1 2 N W t η N (x) .
(2.12)

Then, from (1.18) and (2.12), we have

: z ℓ N (t, x) : = H ℓ (z N (t, x); σ N ) = σ ℓ 2 N H ℓ W t η N (x) . (2.13) Given n ∈ Z 2 , define Γ ℓ (n) by Γ ℓ (n) = {(n 1 , . . . , n ℓ ) ∈ (Z 2 ) ℓ : n 1 + • • • + n ℓ = n}.
Then, for (n 1 , . . . , n ℓ ) ∈ Γ ℓ (n), we have max j |n j | |n|. Thus, it follows from Lemma 2.1 with (2.13) and (2.11) that

: z ℓ N (t) :, e n 2 L 2 (Ω) = σ ℓ N ˆT2 x ×T 2 y e n (x)e n (y) ˆΩ H ℓ W t η N (x) H ℓ W t η N (y) dP dxdy = ℓ! ˆT2 x ×T 2 y [γ N (x -y)] ℓ e n (x -y)dxdy = ℓ! • F[γ ℓ N ](n) = ℓ! Γ ℓ (n) |n j |≤N ℓ j=1 1 n j 2 ρ 1 n 2(1-θ) (2.14)
for any θ > 0. On the other hand, for n = n ′ , we have

ˆΩ : z ℓ N (t) :, e n : z ℓ N (t) :, e n ′ dP = σ ℓ N ˆT2 x ×T 2 y e n (x)e n ′ (y) ˆΩ H ℓ W t η N (x) H ℓ W t η N (y) dP dxdy = ℓ! ˆT2 x ˆT2 y [γ N (x -y)] ℓ e n ′ (x -y)dy e n (x)e n ′ (x)dx = ℓ! • F[γ ℓ N ](n ′ ) ˆT2 x e n (x)e n ′ (x)dx = 0. (2.15)
Hence, given x ∈ T 2 and t ∈ R, it follows from (2.14) and (2.15) that

∇ -ε : z ℓ N (t, x) : L 2 (Ω) = n∈Z 2 n -ε F x : z ℓ N (t) : (n) e in•x L 2 (Ω) ≤ C ℓ n∈Z 2 n -2ε F[γ ℓ N ](n) 1 2 ≤ C ℓ n∈Z 2 n -2(1+ε-θ) 1 2 < ∞, (2.16)
uniformly in N ∈ N, as long as 0 < θ < ε. Fix 2 ≤ q, r < ∞. Then, by Minkowski's integral inequality, Lemma 2.2 (with (2.8)), and (2.16), we have

∇ -ε : z ℓ N : L q T L r x L p (Ω) ≤ ∇ -ε : z ℓ N (t, x) : L p (Ω) L q T L r x ≤ C ℓ p ℓ 2 ∇ -ε : z ℓ N (t, x) : L 2 (Ω) L q T L r x T 1 q p ℓ 2 , (2.17)
for all p ≥ max(q, r). Finally, (2.9) follows from (2.17) and Chebyshev's inequality. A similar computation with Lemma 2.1, (2.11), and Lemma 2.2 shows that the sequence :

z ℓ N : N ∈N is a Cauchy sequence in L p (Ω; L q ([-T, T ]; W -ε,r (T 2 ))).
Remark 2.4. As a corollary to Proposition 2.3, we can show that the tail estimate (2.9) and the convergence of : z ℓ N : to : z ℓ : hold even when q = ∞ and/or r = ∞. This follows from applying Sobolev's inequality (in time and/or space) and using the fact that z solves the linear wave/Klein-Gordon equation. See [START_REF] Burq | Global infinite energy solutions for the cubic wave equation[END_REF]. With this observation, we can easily show that : z ℓ N :, : z ℓ : ∈ C([-T, T ]; W -ε,r (T 2 )) almost surely. See also [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equations[END_REF][START_REF] Oh | Probabilistic global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R 3[END_REF].

Local well-posedness of the Wick ordered NLW

In this section, we present the proof of Theorem 1.2. We combine the deterministic analysis via the Fourier restriction norm method (with the hyperbolic Sobolev spaces) and the stochastic estimate on the Wick ordered monomials : z ℓ : (Proposition 2.3). In the following, we fix ρ > 0.

3.1. Hyperbolic Sobolev spaces and Strichartz estimates. We first recall the hyperbolic Sobolev space X s,b due to Klainerman-Machedon [START_REF] Klainerman | Space-time estimates for null forms and the local existence theorem[END_REF] and Bourgain [START_REF] Bourgain | Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations[END_REF], defined by the norm

u X s,b (R×T 2 ) = n s |τ | -n ρ b u(τ, n) ℓ 2 n L 2 τ (R×Z 2 ) .
For b > 1 2 , we have X s,b ⊂ C(R; H s ). Given an interval I ⊂ R, we define the local-in-time version X s,b (I) as a restriction norm:

u X s,b (I) = inf v X s,b (R×T 2 ) : v| I = u . When I = [-T, T ], we set X s,b T = X s,b (I).
The main deterministic tool for the proof of Theorem 1.2 is the following Strichartz estimates for the linear wave/Klein-Gordon equation. Given 0 ≤ s ≤ 1, we say that a pair (q, r) is s-admissible if 2 < q ≤ ∞, 2 ≤ r < ∞,

1 q + 2 r = 1 -s, and 1 q + 1 2r ≤ 1 4 .
Then, we have the following Strichartz estimates.

Lemma 3.1. Let T ≤ 1. Given 0 ≤ s ≤ 1, let (q, r) be s-admissible. Then, we have

S(t)(φ 0 , φ 1 ) L q T L r x (T 2 ) (φ 0 , φ 1 ) H s (T 2 ) . (3.1)
See Ginibre-Velo [START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF], Lindblad-Sogge [START_REF] Lindblad | On existence and scattering with minimal regularity for semilinear wave equations[END_REF], and Keel-Tao [START_REF] Keel | Endpoint Strichartz estimates[END_REF] for the Strichartz estimates on R d . See also [START_REF] Killip | Blowup behaviour for the nonlinear Klein-Gordon equation[END_REF]. The Strichartz estimates (3.1) on T 2 in Lemma 3.1 follows from those on R 2 and the finite speed of propagation.

When b > 1 2 , the X s,b -spaces enjoy the transference principle. In particular, as a corollary to Lemma 3.1, we obtain the following space-time estimate. See [START_REF] Klainerman | Bilinear estimates and applications to nonlinear wave equations[END_REF][START_REF] Tao | Nonlinear dispersive equations. Local and global analysis[END_REF] for the proof. Lemma 3.2. Let T ≤ 1. Given 0 ≤ s ≤ 1, let (q, r) be s-admissible. Then, for b > 1 2 , we have

u L q T L r x u X s,b T .
Lastly, we state the nonhomogeneous linear estimate. See [START_REF] Ginibre | On the Cauchy problem for the Zakharov system[END_REF].

Lemma 3.3. Let -1 2 < b ′ ≤ 0 ≤ b ≤ b ′ + 1. Then, for T ≤ 1, we have ˆt 0 sin((t -t ′ ) ∇ ρ ) ∇ ρ F (t ′ )dt ′ X s,b T T 1-b+b ′ F X s-1,b ′ T .
3.2. Proof of Theorem 1.2. In the following, we simply consider the case s = s crit + δ with δ ≪ 1. Given T ≤ 1, define Ψ(w) by Then, by duality, we have

Ψ(w)(t) = Ψ ω (w)(t) def = ˆt 0 sin((t -t ′ ) ∇ ρ ) ∇ ρ : (w + z) 2m+1 (t ′ ) : dt ′ . Let b = 1 2 +.
: (w + z) 2m+1 : X s-1,b-1+θ T ≤ 2m+1 ℓ=0 C m,ℓ w 2m+1-ℓ : z ℓ : X s-1,b-1+θ T ≤ 2m+1 ℓ=0 C m,ℓ sup h ℓ ¨1[-T,T ] w 2m+1-ℓ : z ℓ : h ℓ dxdt , (3.3)
for any extension w of w, where the supremum is taken over h ℓ ∈ X 1-s,1-b-θ with h ℓ X 1-s,1-b-θ = 1. By choosing θ > 0 sufficiently small, we have 1 -b -θ = 1 2 -.

• Case 1: m = 1.

In this case, we have s =

s crit + δ = 1 4 + δ. Noting that 12 1+2δ , 3 1-δ is 1 4 + 1 2 δ -admissible, it follows from Lemma 3.2 that ∇ ε w L 12 1+2δ T L 3 1-δ x w X 1 4 + 1 2 δ+ε, 1 2 + w X s, 1 2 +
for any extension w of w, as long as ε ≤ 1 2 δ. By taking an infimum over all the extensions w of w, we obtain inf

w| [-T,T ] =w ∇ ε w L 12 1+2δ T L 3 1-δ x w X s, 1 2 + T . ( 3.4)
On the one hand, noting that 4 1-2δ , 1 δ is 3 4 -3 2 δ -admissible, Hölder's inequality (with T ≤ 1) and Lemma 3.2 yield

∇ ε h ℓ L 4 3-2δ T L 1 δ x ∇ ε h ℓ L 4 1-2δ T L 1 δ x ∇ ε h ℓ X 3 4 -3 2 δ, 1 2 + . (3.5)
On the other hand, applying Hölder's inequality in t and Sobolev's inequality in x, we have

∇ ε h ℓ L 4 3-2δ T L 1 δ x ∇ ε h ℓ L 2 T L 1 δ x ∇ ε h ℓ X 1-2δ,0 . (3.6)
Interpolating (3.5) and (3.6) with sufficiently small θ > 0, we obtain

∇ ε h ℓ L 4 3-2δ T L 1 δ x h ℓ X 3 4 -5 4 δ+ε,1-b-θ h ℓ X 1-s,1-b-θ (3.7)
as long as ε ≤ 1 4 δ. For ℓ = 0, 1, 2, 3, define (q ℓ , r ℓ ) by

1 = (3 -ℓ) 1 + 2δ 12 + 3 -2δ 4 + 1 q ℓ and 1 = (3 -ℓ) 1 -δ 3 + δ + 1 r ℓ .
When ℓ = 0, we have q 0 = r 0 = ∞ and : z 0 : ≡ 1. Then, by fractional Leibniz rule and Hölder's inequality with (3.4) and (3.7), we have inf

w| [-T,T ] =w ¨1[-T,T ] w 3-ℓ : z ℓ : h ℓ dxdt = inf w| [-T,T ] =w ¨1[-T,T ] ∇ ε ( w 3-ℓ h ℓ ) ∇ -ε : z ℓ : dxdt inf w| [-T,T ] =w ∇ ε w 3-ℓ L 12 1+2δ T L 3 1-δ x ∇ ε h ℓ L 4 3-2δ T L 1 δ x ∇ -ε : z ℓ : L q ℓ T L r ℓ x w 3-ℓ X s,b T h ℓ X 1-s,1-b-θ ∇ -ε : z ℓ : L q ℓ T L r ℓ x = w 3-ℓ X s,b T ∇ -ε : z ℓ : L q ℓ T L r ℓ x (3.8)
as long as 0 < ε ≤

Ψ(w) X s,b T T θ 3 ℓ=0 w 3-ℓ X s,b T (3.9)
outside a set of probability < exp -1 T c for some c > 0. Similarly, we have

Ψ(w 1 ) -Ψ(w 2 ) X s,b T T θ 2 ℓ=0 w 1 2-ℓ X s,b T + w 2 2-ℓ X s,b T w 1 -w 2 X s,b T (3.10)
outside a set of probability < exp -1 T c . Therefore, it follows from (3.9) and (3.10) that for each T ≪ 1, there exists a set Ω T with P (Ω c T ) < exp -1 T c such that, for each ω ∈ Ω T , Ψ ω is a contraction on a ball of radius O(1) in X s,b T . • Case 2: m ≥ 2.

In this case, we have s = s crit + δ = 1 -1 m + δ. Define (q, r) by 1 q = 3m -1 3m(2m + 1) + δ 6 and 1 r = 3m + 4 6m(2m + 1) -δ 3 .

Noting that (q, r) is s crit + 1 2 δ -admissible, it follows from Lemma 3.2 that

∇ ε w L q T L r x w X s crit + 1 2 δ+ε, 1 2 + w X s, 1 2 +
for any extension w of w, as long as ε ≤ 1 2 δ. By taking an infimum over all the extensions w of w, we obtain inf

w| [-T,T ] =w ∇ ε w L q T L r x w X s, 1 2 + T . (3.11) Now, define ( q, r) by 1 q = 1 3m - 2m + 1 6 δ and 1 r = 3m -4 6m + 2m + 1 3 δ.
Then, ( q, r) is 1 -s crit -2m+1 2 δ -admissible. On the one hand, by Lemma 3.2, we have

∇ ε h ℓ L q T L r x ∇ ε h ℓ X 1-s crit -2m+1 2 δ, 1 2 + . (3.12)
On the other hand, by Sobolev's inequality, we have

∇ ε h ℓ L q T L r x ∇ ε h ℓ X 1-3m-4 3m -4m+2 3 δ, 1 2 -1 3m + 2m+1 6 δ . (3.13)
Note that the temporal regularity on the right-hand side of (3.13) is less than 1 2 by choosing δ > 0 sufficiently small. Hence, by interpolating (3.12) and (3.13) with sufficiently small θ > 0, we obtain T . This completes the proof of Theorem 1.2.

∇ ε h ℓ L q T L r x h ℓ X 1-s crit -mδ+ε,1-b-θ h ℓ X 1-

Weak universality: Wick ordered NLW as a scaling limit

In this section, we present the proof of Theorem 1.9. We follow closely the argument in [START_REF] Gubinelli | Renormalization of the two-dimensional stochastic nonlinear wave equations[END_REF]. With z ε = z ω ε = S(t)(φ ω ε,0 , φ ω ε,1 ), let us decompose u ε = z ε + w ε as in (1.22). Then, the residual term w ε satisfies

∂ 2 t w ε -∆w ε + ρw ε = F ε (w ε ), (4.1)
where F ε (w ε ) is given by

F ε (w ε ) = ε -3 f (ε(z ε + w ε )) + ε(ε 2 ρ -ρ ε )(z ε + w ε ) = ε -2 {f ′ (0) + ε 2 ρ -ρ ε }(z ε + w ε ) + f (3) (0) 6 (z ε + w ε ) 3 + R ε , (4.2)
where the second equality follows from f (0) = f ′′ (0) = 0 and Taylor's remainder theorem with the remainder term R ε given by

R ε = ε ˆ1 0 (1 -θ) 3 6 f (4) (θε(z ε + w ε ))dθ • (z ε + w ε ) 4 . (4.3)
From (1.30), we see that z ε (t, x) is a mean-zero real-valued Gaussian random variable with variance σ ε = E[z 2 ε (t, x)] ∼ log ε -1 . Note that σ ε is independent of x ∈ T 2 and t ∈ R. Recalling from (2.2) that x 3 = H 3 (x; σ) + 3σx, it follows from (4.2) and (4.3) that

F ε (w ε ) = ε -2 f ′ (0) + ε 2 ρ -ρ ε + 3ε 2 σ ε f (3) (0) 6 (z ε + w ε ) + f (3) (0) 6 H 3 (z ε + w ε ; σ ε ) + R ε .
For each ε > 0, we set ρ ε by

ρ ε = f ′ (0) + ε 2 ρ + ε 2 σ ε f (3) (0) 2
so that the first term on the right-hand side vanishes. Then, by letting λ = f (3) (0) 6 , we obtain

F ε (w ε ) = λH 3 (z ε + w ε ; σ ε ) + R ε def = λ : u 3 ε : +R ε .
From (4.3) and (1.28), we have

|R ε | = ε ˆ1 0 (1 -θ) 3 6 f (4) (θε(z ε + w ε ))dθ • (z ε + w ε ) 4 ε |z ε | + |w ε | M +4 .
In particular, we can write (4.1) as

∂ 2 t w ε -∆w ε + ρw ε = λ 3 ℓ=0 3 ℓ : z ℓ ε : w 3-ℓ ε + O ε |z ε | + |w ε | M +4 . (4.4)
It follows from Proposition 2.3 with (1.30) that

ε z ε M +4 L q T L r x = o ε (1)
almost surely. Then, by proceeding as in Section 3 (where we handle the second term on the right-hand side of (4.4) by applying the argument in Section 3 with 2m + 1 ≥ M + 4), we obtain an a priori bound on w ε , uniformly in ε > 0. Moreover, the local existence time T = T ω can be chosen to be independent of ε > 0.

Let u be the solution to (1.31). In an analogous manner, we can estimate the difference w -w ε , where w = u -z as in (1.22). Together with the almost sure convergence of z ε to z (see Remark 2.4), we see that u ε converges to u in C([-T ω , T ω ]; H s (T 2 )) for s < 0. This completes the proof of Theorem 1.9.

1. 2 .

 2 Gibbs measures and Wick renormalization. With v = ∂ t u, we can write the equation (1.1) in the following Hamiltonian formulation:

2 .

 2 Theorem 1.5. Let M = T 2 , m ∈ N, and ρ > 0. Then, the defocusing Wick ordered NLW (1.24) is almost surely globally well-posed with respect to the Gibbs measure P

  for(1.16) to construct global-in-time weak solutions (without uniqueness) to the Wick ordered NLW(1.24). Moreover, it also allows us to establish invariance of the Gibbs measure P (2m+2) 2 in some mild sense.Theorem 1.7. Let m ∈ N and ρ > 0. Let M be a two-dimensional compact Riemannian manifold without boundary or a bounded domain in R 2 (with the Dirichlet or Neumann boundary condition). In the latter case with the Dirichlet boundary condition, we can also take ρ = 0. Then, there exists a set Σ of full measure with respect to P (2m+2) 2 such that for every φ ∈ Σ, the defocusing Wick ordered NLW (1.24) with initial data distributed according to P (2m+2) 2

  Then, for 0 < θ ≤ 1 -b, by Lemma 3.3, we have Ψ(w) X s,b T T θ : (w + z) 2m+1 : X s-1,b-1+θ T . ℓ : z ℓ : .

w 1 -

 1 s,1-b-θ(3.14) as long as ε ≤ (m -1)δ.Proceeding as before, it follows from Hölder's inequality with(3.11) and(3.14) that¨inf w| [-T,T ] =w 1 [-T,T ] w 2m+1-ℓ : z ℓ : h ℓ dxdt = inf w| [-T,T ] =w ¨1[-T,T ] ∇ ε ( w 2m+1-ℓ h ℓ ) ∇ -ε : z ℓ : dxdt inf w| [-T,T ] =w ∇ ε w 2m+1-ℓ L q T L r x ∇ ε h ℓ L q T L r x ∇ -ε : z ℓ : 1-s,1-b-θ ∇ -ε : z ℓ : 0 < ε ≤ 1 2 δ.Hence, by Proposition 2.3 with (3.2), (3.3), and (3.15), we obtain Ψ(w) X s,w 2 X s,b T outside a set of probability < exp -1 T c for some c > 0. Therefore, for each T ≪ 1, there exists a set Ω T with P (Ω c T ) < exp -1 T c such that, for each ω ∈ Ω T , Ψ ω is a contraction on a ball of radius O(1) in X s,b

Acknowledgements. T.O. is supported by the ERC starting grant no. 637995 "ProbDyn-DispEq". L.T. is supported by the grants "BEKAM" ANR-15-CE40-0001 and "ISDEEC" ANR-16-CE40-0013.