1Introduction and main results

Set \(\Lambda_{\varepsilon} = \varepsilon \mathbb{Z}^2\) and \(\Lambda_{\varepsilon, \ell} = \Lambda_{\varepsilon} / \ell \mathbb{Z}^2\), where the ultraviolet and volume cutoffs are

Here \(L \in \mathbb{N}\) is a fixed constant chosen below. For \(z \in \Lambda_{\varepsilon}\), we define the nearest neighbour Laplacian

For \(\sigma \in (0, 2]\), the operator \(\mathcal{A}_{\varepsilon} := (I - \Delta_{\varepsilon})^{\sigma / 2}\) is defined on \(\Lambda_{\varepsilon}\) as the Fourier multiplier. Write \(\nu_{\varepsilon, \ell}\) for the Gaussian measure on \(\mathbb{R}^{\Lambda_{\varepsilon, \ell}}\) with covariance \(\mathcal{A}_{\varepsilon}^{- 1}\). The interacting fractional \(\Phi^4_2\) measure on \(\mathbb{R}^{\Lambda_{\varepsilon, \ell}}\) is

We study the above measure in the regime in which the so-called Wick renormalisation is sufficient to control the continuum limit \(\varepsilon \searrow 0\). Hence, the counterterm coincides with

where \(G_{\varepsilon}\) is the kernel of \(\mathcal{A}_{\varepsilon}^{- 1}\) on \(\Lambda_{\varepsilon}\). Let \(A \subset \Lambda_{\varepsilon, \ell}\). An observable \(F = F_{\varepsilon, \ell} \in C^1\) is called \(A\)-supported if \(\text{supp} \nabla F (\phi) \subseteq A\), where \(\nabla F\) is the gradient relative to \(\langle \cdot, \cdot \rangle\). We denote by \(d_{\ell}\) the \(\ell^{\infty}\) distance on the torus \(\mathbb{T}_{\ell}^2 =\mathbb{R}^2 / (\ell \mathbb{Z}^2)\) and write \(d_{\ell} (A, B)\) for the distance between the \(\ell\)-periodic sets \(A, B \subset \mathbb{T}_{\ell}^2\).

The assumption that \(\lambda > 0\) is small enough corresponds to the high-temperature regime. The case \(\sigma = 2\) corresponds to the standard \(\Phi^4_2\) model. The exponential decay relies on our choice of the fractional operator \((1 - \Delta_{\varepsilon})^{\sigma / 2}\). For \(\sigma < 2\), the alternative natural choice \(1 + (- \Delta_{\varepsilon})^{\sigma / 2}\) gives a Gaussian measure whose covariance decays only polynomially.

The proof technique does not use the scalar nature of the model and extends to the vectorial model as briefly discussed in Appendix C. By a straightforward adaptation of the technique developed in [DGR26], after a suitable extension to \(\mathscr{S}' (\mathbb{R}^2)\), the family of measures \((\mu_{\varepsilon, \ell})_{\varepsilon, \ell}\) is tight in the entire subcritical regime \(\sigma \in (1, 2]\), for a suitable choice of counterterm that coincides with (1.3) when \(\sigma \in (4 / 3, 2]\). The bound (1.4) holds for any accumulation point for \(\sigma \in (4 / 3, 2]\). The extension for \(\sigma \in (1, 4 / 3]\), that is beyond the Wick-ordering regime, requires additional technical arguments, which we leave for future work.

The proof of the above theorem is presented in Section 6. It uses the Helffer–Sjöstrand covariance representation [HS94] combined with the analysis of the parabolic stochastic quantisation equation of the model,

where \(W\) is standard Brownian motion in \(L^2 = L^2 (\Lambda_{\varepsilon, \ell}, \varepsilon^2 \text{counting measure})\) extended periodically to \(\Lambda_{\varepsilon}\). In particular,

At fixed cutoffs (1.5) is a globally well-posed finite-dimensional SDE. Its unique invariant measure is (1.2), and the Markov semigroup \(P_t\) is reversible in \(L^2 (\mu_{\varepsilon, \ell})\). Write \(t \mapsto \Phi_t^{\phi}\) for the solution of (1.5) started from \(\phi\) at time \(0\). For an arbitrary initial time \(s\), we write \(t \mapsto \Phi_{s, t}^{\phi}\) for the solution satisfying \(\Phi_{s, s}^{\phi} = \phi\). Its derivative with respect to the initial condition is

Note that \(t \mapsto U_{s, t}^{\phi} f\) solves

We divide the lattice \(\Lambda_{\varepsilon, \ell}\) into blocks of size \(L\) and denote by \(\chi_{\alpha}\) a bump function supported in a block centred at a lattice point \(\alpha\); see Section 2.1. Our main technical result is the following contractive estimate for the linearised dynamic.

Proof. Proposition 5.2 proves the hypothesis (2.14) of Proposition 2.2, which then implies the stated bound.\(\Box\)

The bound (1.7) shows that a perturbation of the initial condition decays in time and has exponentially small influence on distant spatial regions uniformly in the cutoffs. This estimate extends to the infinite-volume continuum dynamic and yields exponential synchronisation of solutions driven by the same noise. Consequently, the dynamic is ergodic and admits a unique invariant measure, with every solution converging exponentially fast to the unique stationary solution. The invariant measure is invariant under all Euclidean transformations and satisfies reflection positivity. We give a self-contained proof of the results stated as theorems. The aforementioned conclusions about the limiting dynamic and its invariant measure are not proved here, but follow easily along the lines of [DHYZ25; GH21]. The proof of synchronisation by noise is outlined in Appendix D.

Proof idea.

Related literature.

Comparison with [DHYZ25].

Notation.

The \(L^2\) inner product and \(L^p\) norms are over \(\Lambda_{\varepsilon, \ell}\). In particular,

We stress that sums in inner products, norms and quadratic forms always run over \(\Lambda_{\varepsilon, \ell}\), whereas the sum in the definition of the convolution is over \(\Lambda_{\varepsilon}\).

Acknowledgements.

AI disclosure statement.

2Spatial propagation

2.1Block decomposition and random walk representation

We begin by studying how a perturbation of the initial data propagates under the linearised dynamic (1.6). We divide the spatial lattice into overlapping blocks of size \(L\), indexed by their centres \(\alpha\). For each block, we define the local \(L^2\) energy \(e_{\alpha}\) of the linearised solution inside a block \(\alpha\). The kinetic operator \(\mathcal{A}_{\varepsilon}\) transfers energy between blocks, and controlling this transfer is essential for establishing decay of correlations. The main result shows that the transfer of energy between blocks is dominated by an auxiliary random walk with exponentially decaying jump rates. This reduces the problem of studying spatial propagation to estimating the auxiliary evolution (2.7) on the lattice \(\mathscr{B}_{\ell}\) of block centres.

Fix a nonnegative function \(\chi \in C_c^{\infty} (\mathbb{R}^2)\), supported in \([- 1, 1]^2\) and strictly positive on \([- \tfrac{1}{2}, \tfrac{1}{2}]^2\). The parameter \(L > 16\) introduced below is fixed sufficiently big, depending only on \(\sigma\) and \(\chi\), such that the bounds (2.5), (2.6), (A.3) and (A.4) below hold true. Subsequently, the constant \(\lambda_0 > 0\) is fixed sufficiently small. In what follows, constants \(C \in (0, \infty)\) may change from line to line, and are always independent of the cutoffs, the coupling constant \(\lambda \in (0, \lambda_0]\), and block indices.

For torus size \(\ell = M L\), let

be the lattice of block centres. We reserve the symbols \(\alpha, \beta, \bar{\alpha}, \bar{\beta} \in \mathscr{B}_{\ell}\) for block labels. For \(z \in \Lambda_{\varepsilon, \ell}\), set

By our assumptions on \(\chi\), the denominator is strictly positive. Thus, \(\chi_{\alpha}\) is smooth, and

The support assumption gives

Put

In two dimensions, we have

Let \(q_{\varepsilon} (z, y)\) be the kernel introduced in Lemma A.1 such that

For any lattice field \(v\) define the local energy and the localisation error

A direct computation yields

For each deterministic initial condition \(\phi\), define

Note that the local Rayleigh quotient \(\mathcal{R}^{\phi}_{\alpha} (t)\) describes the smallest value of the quadratic form of the random Schrödinger operator

appearing in the linearised equation (1.6), on functions supported in a given block \(\alpha\).

Proof. Set

By symmetry and Cauchy–Schwarz,

The bound \(| \chi_{\alpha} (z) - \chi_{\alpha} (y) | \leqslant CL^{- 1} d_{\ell} (z, y)\) and (A.2) imply \(k_{\alpha} (z) \leqslant CL^{- 2}\). If \(d_{\ell} (z, \alpha) > L\), then \(\chi_{\alpha} (z) = 0\) and \(\chi_{\alpha} (y) \ne 0\) can only hold if \(d_{\ell} (z, y) \geqslant d_{\ell} (z, \alpha) - L\). Thus,

where the last inequality holds for \(L \geqslant 8\), after increasing \(C\).

If \(\chi_{\beta} (z) \ne 0\), then \(d_{\ell} (\alpha, \beta) \leqslant d_{\ell} (z, \alpha) + L\) by the triangle inequality. Using \(\sum_{\beta} \chi_{\beta}^2 = 1\), we obtain

It follows from (2.8) that

We have

Let \(\mathcal{J}\) be a conservative Markov generator with rates \(j_{\alpha \beta} = a_{\alpha \beta}\) for \(\beta \ne \alpha\). Then

Choosing \(L\) sufficiently large and using (2.10) we obtain (2.5) and (2.6).

Finally, differentiating \(e_{\alpha} (v)\) in time and using (1.6), the localisation identity (2.3), the definition of the Rayleigh quotient (2.4), and (2.5) gives (2.7).\(\Box\)

2.2An abstract propagation criterion

The local energy inequality (2.7) contains two competing terms: a dissipative term \(- e_{\alpha}\), which produces exponential decay, and a random potential \(\hat{\mathcal{R}}^{\phi}_{\alpha}\), which may produce a local instability. The Feynman–Kac representation allows us to express the resulting energy bound as an average over paths of an auxiliary random walk \(\Pi\) on spatial blocks. For \(T \geqslant 0\), \(k \geqslant 1\), and a piecewise-constant path \(\pi : [1, 1 + T] \to \mathscr{B}_{\ell}\), define

and

Along each path, the random potential in (1.6) produces an exponential factor determined by the accumulated loss of dissipation. To preserve exponential decay, we formulate the criterion (2.14), which allows a small cost \(\mathcal{N} (\pi) / 8\) for each block visited by the walk during a unit time interval. It also permits an additional, potentially big, contribution \(\mathcal{T} (\pi)\) arising from the coming-down-from-infinity estimate for the nonlinear dynamic (see the first term on the RHS of (4.4)). The distinction between these two contributions matters. Since the jump rates of the block walk are uniformly bounded, the exponential moments of \(\mathcal{N} (\pi)\) grow at most exponentially in \(T\). Thus, their growth must be sufficiently small to be dominated by the deterministic dissipation. In contrast, the exponentially decaying weights in \(\mathcal{T} (\pi)\) imply that its exponential moments are bounded uniformly in \(T\), even when multiplied by an arbitrarily large fixed constant. Thus the contribution involving \(\mathcal{T} (\pi)\) changes only the multiplicative constant in the final estimate, whereas the contribution involving \(\mathcal{N} (\pi)\) reduces the exponential decay rate. Note also that the naive bounds

cannot be used directly to verify our criterion as \(C_{\varepsilon, \ell} \rightarrow \infty\) as \(\varepsilon \searrow 0\). To show (2.14) one has to exhibit stochastic cancellations in the local Rayleigh quotient \(\mathcal{R}^{\phi}_{\alpha} (t)\). These cancellations will be analysed in subsequent sections.

We also introduce the path functional

measuring the accumulated loss of dissipation, where the supremum in the definition of \(\mathcal{O}_{\ast} (\pi)\) runs over deterministic initial conditions \(\phi\). Since \(\mathcal{O}^{\phi} (\pi)\) depends continuously on \(\phi\) and the finite dimensional state space is separable, \(\mathcal{O}_{\ast} (\pi)\) is measurable. The terminal time \(T + 1\) is determined by the domain of \(\pi\). Let \(\Pi\) be the continuous-time random walk on \(\mathscr{B}_{\ell}\) generated by \(\mathcal{J}\), independent of the noise, and denote by \(\boldsymbol{E}_{\beta}^{\Pi}\) the expectation conditioned on \(\Pi_0 = \beta\). Define the uniform exponential jump cost

where \(j_{\alpha \beta}\) is the jump kernel of \(\mathcal{J}\) from (2.5).

Proof. Let \(v^{\phi}\) solve the linearised equation (1.6) along \(\Phi^{\phi}\) on \([1, 1 + T]\), with \(v_1^{\phi} = \chi_{\alpha} f\), and put \(e^{\phi}_{\alpha} (t) = \| \chi_{\alpha} v^{\phi}_t \|_2^2\). The positive-semigroup comparison applied to (2.7) and backward Feynman–Kac formula yield

The inequality holds pathwise for every initial condition. Since \(v_1^{\phi} = \chi_{\alpha} f\), we have

Define the random backward path \(\pi (r) = \Pi_{1 + T - r}\). Taking the supremum over initial conditions in (2.15) gives

Taking the supremum over \(\|f\|_2 \leqslant 1\) therefore yields

Hence by (2.14), we obtain

Lemma A.3, applied with \(a = C\) and \(b = 1 / 8\), now gives

Finally, by (2.6),

This proves (1.7).\(\Box\)

3Stochastic enhancement and remainder dynamic

The energy estimates discussed in Section 4 will be applied to a decomposition of the stochastic quantisation solution into an explicit singular stochastic part and a more regular remainder. We first introduce the stationary Gaussian field \(X\) and its Wick powers. For \(\sigma > 3 / 2\), the Wick cube \(X^{: 3 :}\) is sufficiently regular to be treated directly as a forcing term in the energy estimates, and one may simply use the decomposition \(\Phi = X + \bar{\Psi}\). To cover the entire range \(\sigma > 4 / 3\), we introduce the second order correction \(Y\) to remove \(X^{: 3 :}\) from the remainder equation.

Let \(W\) be the two sided Wiener process. Define stationary stochastic convolutions

where \(C_{\varepsilon, \ell}\) is the counterterm introduced in (1.3). Note that \(C_{\varepsilon, \ell} =\mathbb{E} [X_t (z)^2]\). Let \(p_t^{\varepsilon}\) be the infinite-lattice convolution kernel of \(e^{- t\mathcal{A}_{\varepsilon}}\). We have

Put \(Z := X - \lambda Y\) and

The fields \(V\) and \(H\) play the role of the random potential and the forcing term in the equation for the remainder \(\Psi := \Phi - Z\).

For a deterministic initial condition \(\phi\) and a control \(u\), let \(\Phi^{\phi, u}\) denote the solution of the stochastic quantisation equation with the additional drift \(\sqrt{2} u\) and the initial condition \(\Phi_0^{\phi, u} = \phi\). Define \(\Psi^{\phi, u} : = \Phi^{\phi, u} - Z\). We suppress the dependence on the initial condition in the notation when \(\phi\) is fixed, and write \(\Phi^u = \Phi^{\phi, u}\) and \(\Psi^u = \Psi^{\phi, u}\).

Proof. Expand \((Z + \Psi^u)^3 - 3 C_{\varepsilon, \ell} (Z + \Psi^u)\) using \(Z = X - \lambda Y\), and cancel the cubic Wick term using the equation for \(Y\).\(\Box\)

Fix \(0 < \delta < (3 \sigma - 4) / 12\) and set

Then \(\sigma - 4 \eta > 0\) and \(2 \eta < \sigma / 2\). Choose

and define

The corresponding weighted Besov norms are defined in Appendix B. The constant \(p\) plays the role of the exponent of the size of the enhancement \(\mathfrak{A}_{\alpha} (t)\) introduced below, and is chosen sufficiently large to absorb the powers of \(\mathfrak{A}_{\alpha} (t)\) arising later in the energy and Rayleigh-quotient estimates in Section 4. The choice of \(\theta\) ensures \(8 \theta < 2\), which guarantees the spatial integrability of \(w_{\alpha}^2 \omega_{\alpha}^{- 8}\) used in Lemma B.5.

Note that \(\mathfrak{A}_{\alpha} (t)\) does not depend on \(\lambda\). By Lemma B.3, there is a deterministic constant \(C\) such that for all \(\alpha \in \mathscr{B}_{\ell}\), \(t \in \mathbb{R}\), and \(0 < \lambda \leqslant 1\),

Proof. The result can be proved by adapting the proof technique from Section 6 of [ZZ18], where analogous stochastic estimates are proved in the more complicated setting of the \(\Phi^4_3\) model.\(\Box\)

4Pathwise energy and Rayleigh estimates

4.1Weighted energy inequality

To establish the exponential-moment bound (2.14) required by Proposition 2.2, we shall apply in Section 5 the Boué–Dupuis variational formula. In this section, we derive suitable pathwise energy estimates for solutions of the stochastic quantisation equation (1.5) with an additional drift \(u\), which will play the role of the control in the Boué–Dupuis representation.

For \(q \geqslant 1\) define

Let \(u \in L^2 ([0, t] ; L^2)\) be a control. Recall that the controlled dynamic satisfies (1.5) with an additional drift \(\sqrt{2} u\) and the initial condition \(\Phi_0^u = \phi\). Define

where \(\Psi^u = \Phi^u - Z\).

Proof. Write

Testing (3.1) against \(2 w_{\alpha}^2 \Psi^u\) and applying Lemmas A.2 and B.5 gives

Since \(K \geqslant E\) and \(2 \sqrt{2} | \langle w_{\alpha} \Psi^u, w_{\alpha} u \rangle | \leqslant \frac{1}{4} E + C \|w_{\alpha} u\|_2^2\), we obtain

Finally, (A.4) gives \(L^{- 2} E^2 \leqslant 2 P\). Hence

This proves (4.2).\(\Box\)

4.2Coming down from infinity and dissipation

The energy inequality obtained above depends on the initial condition, whereas the estimate stated in Theorem 1.2 is uniform in the initial data. To obtain the desired bound, we use the coming-down-from-infinity estimate (4.7) below to bound the initial energy in terms of the forcing at earlier times, with a weight decaying exponentially in time.

Proof. Let \(C\) be as in (4.2) and put

Since \(\partial_r h + h / 2\) coincides with the forcing in (4.2) and \(h \geqslant 0\), setting \(g = (\mathcal{E}_{\alpha}^u - h)_+\), we arrive at

where we used \(\mathcal{E}_{\alpha}^u \geqslant g\). If \(g > 0\), the inverse \(1 / g\) satisfies

Integrating this inequality gives

If \(g\) vanishes at some point in time, it must be identically zero by (4.6). Consequently,

Thus, independently of the initial energy,

The bound (4.4) follows by integrating (4.2) over \(I_k^T\), discarding the nonnegative terms, and bounding the energy at \(k\) by (4.7), using \((e^{k / 2} - 1)^{- 1} \leqslant Ce^{- k / 2}\) for \(k \geqslant 1\). The kernel bounds follow by elementary estimates.\(\Box\)

4.3Local Rayleigh-quotient bound

For a reference solution \(\Phi^u = Z + \Psi^u\) starting from the initial condition \(\phi\) and driven by the noise shifted by \(u\), let \(\mathcal{R}_{\alpha}^{\phi, u}\) be defined by (2.4) with \(\Phi^{\phi}\) replaced by \(\Phi^{\phi, u}\), and set \(\hat{\mathcal{R}}_{\alpha}^{\phi, u} = (1 -\mathcal{R}_{\alpha}^{\phi, u})_+\). The main difficulty in estimating the local Rayleigh quotient is the absence of a uniform lower bound on the random potential due to the presence of the counterterm. The decomposition \(\Phi^u = Z + \Psi^u\) of the nonlinear solution separates this potential into a non-negative term quadratic in the remainder and two potentially negative terms involving the stochastic coefficients,

We estimate the last two terms using the weighted Besov estimates, which allow us to bound their contributions by the kinetic term and \(\mathcal{D}_{\alpha}^u\). As a result, we can bound \(\hat{\mathcal{R}}_{\alpha}^{\phi, u} (t)\) entirely in terms of the dissipation \(\mathcal{D}_{\alpha}^u\) and the stochastic enhancement.

Proof. By (3.3) and Lemma B.4, we have

Since \(\| \varphi \|_2 = 1\), interpolation between Sobolev spaces gives

with

Hence, by Young's inequality

By the assumptions on \(\delta\) and \(p\) from Section 3,

Since \(\mathfrak{A}_{\alpha} \geqslant 1\), \(\| \varphi \|_{H^{\sigma / 2}}^2 \lesssim 1 + \langle \varphi, (\mathcal{A}_{\varepsilon} - I) \varphi \rangle\) and \(\|w_{\alpha} \Psi^u \|_{H^{\sigma / 2}}^2 \leqslant \mathcal{D}_{\alpha}^u\), the above estimates imply (4.9). Finally, using (2.4) and discarding the nonnegative first term in (4.8), we obtain (4.10).\(\Box\)

5Path estimates and Boué–Dupuis

5.1Bounds along block paths

For a path \(\pi : [1, 1 + T] \to \mathscr{B}_{\ell}\) and a control \(u\), define \(\mathcal{O}^{\phi, u} (\pi)\) and \(\mathcal{O}^u_{\ast} (\pi)\) by replacing \(\hat{\mathcal{R}}^{\phi}_{\alpha}\) with \(\hat{\mathcal{R}}_{\alpha}^{\phi, u}\) in (2.12). Explicitly, \(\mathcal{O}_{\ast}^u (\pi) := \sup_{\phi \in L^2} \mathcal{O}^{\phi, u} (\pi)\). In this section, we estimate the accumulated loss of dissipation \(\mathcal{O}_{\ast}^u (\pi)\) along a fixed path \(\pi\) by combining the local Rayleigh-quotient bound with the energy estimates for the nonlinear dynamic. The estimate is uniform in the initial condition and the cutoffs and provides the main input for the application of the Boué–Dupuis formula in the next subsection.

Proof. For \(r \in I_k^T\), \(\pi (r) \in \textsf{V}_k (\pi)\). Hence (4.10) yields

The contribution from the first term on the RHS of (4.4) sums to \(CL^2 \mathcal{T} (\pi)\). As \(K_k^T \geqslant \textbf{1}_{I_k^T}\) and \(\lambda \leqslant 1\), the enhancement terms are bounded by \(C \lambda \mathfrak{M} (\pi)\), where

By Proposition 3.3 and (4.5), \(\mathbb{E}\mathfrak{M} (\pi) \leqslant C\mathcal{N} (\pi)\). The contribution from the term on the RHS of (4.4) involving the control \(u\) is bounded independently of the number of blocks visited by the path. Indeed, by (2.1), we have

Summing over \(k\) and using (4.5), we bound the total control contribution by \(C \lambda \int_0^{1 + T} \|u (r)\|_2^2 \hspace{0.17em} \text{d} r\). Since the estimates above hold pathwise with a right-hand side independent of the initial condition, taking the supremum over all initial conditions gives (5.1).\(\Box\)

5.2Exponential path bound

We finally establish the exponential-moment estimate in the propagation criterion formulated in Proposition 2.2. This finishes the proof of the decay estimate for the linearised evolution stated in Theorem 1.2.

Proof. Since \(\mathcal{A}_{\varepsilon} \geqslant I\) and \(3 \lambda (\Phi^2 - C_{\varepsilon, \ell}) \geqslant - 3 \lambda C_{\varepsilon, \ell}\), we have \(0 \leqslant \mathcal{O}^{\phi, u} (\pi) \leqslant 3 \lambda C_{\varepsilon, \ell} T\). For deterministic initial data the functional \(\mathcal{O}^{\phi} (\pi)\) depends only on the Wiener process after time zero. The same properties are true for \(\mathcal{O}_{\ast} (\pi)\). Thus the Boué–Dupuis formula [BD98] applies to this bounded functional and yields

The supremum is over controls \(u\) with finite expected \(L^2\) energy progressively measurable with respect to the filtration generated by the Wiener process after time zero. To bound the shifted functional, we use the decomposition \(\Phi^{\phi, u} = Z + \Psi^{\phi, u}\) and use the enhancement from Section 3, which depends on the original, unshifted noise also before time zero. Note that the use of the noise before time zero plays an auxiliary role and does not affect the variational formulation. To finish the proof insert (5.1) and fix \(\lambda_0\) sufficiently small that \(2 C \lambda_0 \leqslant 1 / 8\).\(\Box\)

6Proof of the mass gap

To prove Theorem 1.1 we combine the estimate for the linearised dynamic from Theorem 1.2 with the Helffer–Sjöstrand [HS94] representation of the covariance. The estimate for the linearised evolution uses the coming-down-from-infinity property and becomes uniform in the initial condition only after a positive time. To exploit this estimate in the covariance representation below, we use stationarity to shift the tangent evolution from \([0, T]\) to \([1, 1 + T]\).

Proof. Let \(P_T\) be the Markov semigroup associated to (1.5) and write \(\mu = \mu_{\varepsilon, \ell}\) for the invariant measure. Reversibility and integration by parts give

Since the finite-dimensional dynamic (1.5) is ergodic, \(P_T G \to \int G \hspace{0.17em} \text{d} \mu\) in \(L^2 (\mu)\) as \(T \to \infty\). Integrating the preceding identity yields

By the chain rule,

Combining the last two identities and using stationarity to shift the time proves (6.1).\(\Box\)

Proof of Theorem 1.1. Let \(\Phi\) be stationary with fixed-time law \(\mu_{\varepsilon, \ell}\), and set

By the partition of unity,

Set

Cauchy–Schwarz and Theorem 1.2, applied conditionally on \(\Phi_0\) with initial data \(\chi_{\alpha} f\) for the linearised dynamic, give

As \(A\) and \(B\) have diameter at most \(R\), there are only finitely many \(\alpha\) and \(\beta\) such that \(\text{supp} \chi_{\alpha} \cap A \neq \emptyset\) and \(\text{supp} \chi_{\beta} \cap B \neq \emptyset\). Moreover, for such indices,

Consequently,

Integrating over \(T \geqslant 0\) and using (6.1) proves (1.4).\(\Box\)

Appendix ARandom-walk estimates

In this appendix we collect estimates for the discrete random walk. We first represent the nonlocal operator \(I -\mathcal{A}_{\varepsilon}\) as the generator of a symmetric random walk and establish exponential moment bounds for its jump kernel. These bounds are then used to control errors in the spatial localisation. Finally, we estimate exponential moments of the number of blocks visited by the random walk. The latter estimate is the key ingredient in the proof of the abstract propagation criterion (2.14).

Proof. For \(\sigma = 2\), the kernel is given, for \(y \ne z\), by

where \(\text{e}_1 = (1, 0)\) and \(\text{e}_2 = (0, 1)\). The claimed properties follow directly.

Suppose \(\sigma < 2\). Let \(S_s\) be the continuous-time nearest-neighbour walk on \(\Lambda_{\varepsilon}\), started at \(0\), with generator \(\Delta_{\varepsilon}\), and let \(\bar{S}_s\) denote its projection onto \(\Lambda_{\varepsilon, \ell}\). The identity

gives the desired generator representation with \(q_{\varepsilon} (z, z) = 0\) and, for \(y \ne z\),

In particular, \(q_{\varepsilon}\) is nonnegative and symmetric. Let \(\varsigma \in \{- 1, 1\}^2\). Since \(z \mapsto e^{\vartheta \varsigma \cdot z}\) is an eigenfunction of \(\Delta_{\varepsilon}\) with eigenvalue \(4 \varepsilon^{- 2} (\cosh (\varepsilon \vartheta) - 1)\), we obtain

Differentiating twice in \(\vartheta\), summing over \(\varsigma\), and using \(0 < \varepsilon \leqslant 1\), we obtain

Since \(d_{\ell} (0, \bar{S}_s) \leqslant |S_s |\), it follows that

The integral is finite because \(\sigma < 2\) and \(c < 1\). This proves (A.2).\(\Box\)

Proof. Using the fact that \(\rho \mapsto \log (1 + \rho / L)\) is Lipschitz on \([0, \infty)\) with Lipschitz constant \(L^{- 1}\), as well as the triangle inequality for \(d_{\ell}\), we obtain \(| \log w_{\alpha} (z) - \log w_{\alpha} (y) | \leqslant 2 d_{\ell} (z, y) / L\). Consequently, for \(L \geqslant 8\), we have

By (A.1) we have

Therefore, (A.2) together with \(2 |ab| \leqslant a^2 + b^2\) give

Taking \(L\) sufficiently large proves (A.3).

For the second inequality, we note that

Consequently, Cauchy–Schwarz gives

which proves (A.4).\(\Box\)

Proof. Set \(c_{\ast} = \sum_{k \geqslant 1} e^{- k / 2} < \infty\). For almost every \(r \in [0, T]\), let \(k (r)\) be fixed by \(1 + T - r \in I_{k (r)}^T\). In the following, sums over \(s < r \leqslant T\) run over the times when the walk \(\Pi\) jumps. We have

where \(N^{\Pi}_T\) is the number of jumps of \(\Pi\) during \([0, T]\). Moreover, the triangle inequality and \(\Pi_0 = \beta\) give

Combining the preceding estimates yields

where

For \(0 \leqslant s \leqslant T\), set

where \(\boldsymbol{E}_{s, \bar{\alpha}}^{\Pi}\) denotes the expectation for the walk started at \(\bar{\alpha}\) at time \(s\). By conditioning on the first jump, \(h\) satisfies, for almost every \(s \in [0, T]\),

Set

The spatially constant function \(v (s) = \exp \left( \int_s^T m (r) \hspace{0.17em} \text{d} r \right)\) is a supersolution of the above backward equation, with the same terminal condition. Since the off-diagonal coefficients are nonnegative, comparison gives \(h (0, \beta) \leqslant v (0)\), proving

By (2.13), (2.6), and \(e^{a \rho} - 1 \leqslant ae^a \rho\) for \(0 \leqslant \rho \leqslant 1\),

We have

Consequently, by (A.7) we arrive at

Combining this with (A.6), we obtain (A.5) with \(C_{a, b} = \exp (b + ac_{\ast} + \tfrac{1}{2} ae^{a + b} c_{\ast})\).\(\Box\)

Appendix BWeighted lattice Besov estimates

In this appendix we collect deterministic estimates in weighted Besov spaces on the lattice \(\Lambda_{\varepsilon}\) used to control products involving the singular stochastic terms and the nonlinear remainder.

Note that by Minkowski's inequality and \(\omega (z)^m / \omega (z - x)^m \leqslant C (1 + |x| / L)^{m \theta}\), we have

The Fourier symbol of \(- \Delta_{\varepsilon}\) satisfies

on the Brillouin zone. Therefore, for \(\rho \in \mathbb{R}\),

Proof. The result follows by Besov duality from [GH21, Lemmas A.1–A.2]. Note that the periodic weights \(\omega_{\alpha}^m\) may be replaced with smooth periodic weights, without changing the norms.\(\Box\)

Proof. Apply the Bony decomposition of the product and use [MP19, Lemma 4.2].\(\Box\)

Proof. Write \(w = w_{\alpha}\) and \(\omega = \omega_{\alpha}\). Since \(\text{supp} \varphi \subset B_{\alpha}\), the weights and their inverses are bounded and Lipschitz on a fixed neighbourhood of \(\text{supp} \varphi\). Hence, the Sobolev multiplier estimate yields

By (B.5) and [GH21, Lemma A.7],

This proves (B.7). For the second estimate, first note that the paraproduct bounds of [MP19, Lemma 4.2], together with the Sobolev embeddings, yield

Apply this with \(f_1 = w \psi\), \(f_2 = f_3 = (w \omega)^{- 1 / 2} \varphi\) and \(s_1 = \sigma / 2\), \(s_2 = s_3 = \eta + \delta / 2\). The bound (B.8) follows then by (B.5) and (B.9).\(\Box\)

Proof. Write \(w = w_{\alpha}\) and \(\omega = \omega_{\alpha}\). Since \(8 \theta < 2\), we have

Therefore,

We use the weighted Besov estimates of [GH21, Appendix A], replacing our periodic weight by comparable smooth periodic weights. Observe also that by (3.2) and the choice of \(\delta\),

By [GH21, Lemmas A.1, A.3 and A.4], we obtain

Hence, by (B.5) and [GH21, Lemma A.7], we have

Similarly, the quadratic estimate of [GH21, Lemma A.7] gives

For the linear term, weighted Besov embedding and Hölder's inequality give

Combining these estimates and applying Young's inequality, we get

The result follows now from (3.3), since \(p \geqslant 8\).\(\Box\)

Appendix CExtension to the \(O (N)\)-invariant model

We briefly discuss the extension to the \(\Phi^4\) model with fields valued in \(\mathbb{R}^N\), for fixed \(N \in \mathbb{N}\). The interaction potential is

The corresponding stochastic quantisation equation is

where the components of \(W = (W^1, \ldots, W^N)\) are independent.

Let us explain how to adapt the construction of the stochastic objects from Section 3. We define \(X\) to be the stationary vector-valued Gaussian convolution and set

The scalar Wick square \(X^{: 2 :}\) is replaced by the matrix \(B_X = X \otimes X - C_{\varepsilon, \ell} I_N\). Summing up, the required enhancement consists of

with the same weighted Besov regularities as \(X\), \(X^{: 2 :}\), \(Y\), and \(X^{: 2 :} Y\) in the scalar case.

To see why the deterministic estimates have the same structure, put \(\Phi^u = Z + \Psi^u\) and \(B_Z = Z \otimes Z - C_{\varepsilon, \ell} I_N\), where \(I_N\) is the identity matrix. Expansion of the original cubic nonlinearity produces the coercive term \(\lambda | \Psi^u |^2 \Psi^u\), quadratic in \(\Psi^u\) term involving \(Z\), a linear term involving \(B_Z\), and a forcing coinciding with a linear combination of terms of the types \(B_X Y\), \(XY^2\), and \(Y^3\). In order to obtain a desired energy bound we can apply componentwise the product estimates in Lemma B.5.

The main point in the local Rayleigh estimate is that the Hessian of the interaction is matrix-valued but has the decomposition

where the first two terms have quadratic form \(| \Psi^u |^2 |v|^2 + 2 (\Psi^u \cdot v)^2 \geqslant 0\). The remaining terms are controlled componentwise by the quadratic and trilinear bounds of Lemma B.4. This gives the analogue of Proposition 4.3, with the vector-valued local Rayleigh quotient defined using the symmetric quadratic form above.

Finally, the localisation and block-transfer argument acts on the scalar energies \(e_{\alpha} = \| \chi_{\alpha} v\|_2^2\) and requires no positivity of a matrix-valued propagator. The block walk, Boué–Dupuis argument (with vector-valued controls), and covariance representation do not require any modifications. Thus, the proofs of Theorems 1.1 and 1.2 require only very minor modifications.

Appendix DSynchronisation by noise

Let us explain how to conclude almost sure exponential synchronisation of solutions of the infinite volume continuum equation driven by the same noise. This requires a minor extension of Theorem 1.2. Recall that \(\mathfrak{S}_1 = \{\Phi_1^{\phi} : \phi \in L^2 \}\). We claim that the estimate (1.7) is also valid when the supremum over \(\psi \in \mathfrak{S}_1\) is replaced with the supremum over its closed convex hull \(\overline{\text{co}} (\mathfrak{S}_1)\).

Let us first show that (4.4) holds uniformly over \(\psi \in \overline{\text{co}} (\mathfrak{S}_1)\). By convexity of the squared \(L^2\) norm,

where \(\mathfrak{S}_1^u := \{\Phi_1^{\phi, u} : \phi \in L^2 \}\) and \(Z\) is the process introduced in Section 3. The RHS is bounded by the coming-down estimate (4.7). Consequently, applying the energy inequality (4.2) to solutions restarted at time one with the initial condition \(\psi \in \overline{\text{co}} (\mathfrak{S}_1^u)\) yields (4.4). Thus, the proof of Lemma 5.1 applies uniformly over \(\overline{\text{co}} (\mathfrak{S}_1^u)\). Under a Cameron–Martin shift, \(\mathfrak{S}_1\) is transformed into \(\mathfrak{S}_1^u\), so Proposition 5.2 and the argument of Proposition 2.2 prove (1.7) with the supremum over \(\psi \in \overline{\text{co}} (\mathfrak{S}_1)\). This proves the convex hull extension of Theorem 1.2.

To establish synchronisation, we study the random diameter of the solution set at time \(t\),

where the supremum runs over all initial conditions. For arbitrary initial conditions \(\phi_1, \phi_2\), write \(\psi_i = \Phi_1^{\phi_i}\). Differentiability of the solution map gives

Since \((1 - a) \psi_2 + a \psi_1 \in \overline{\text{co}} (\mathfrak{S}_1)\), we obtain

By (4.7), (2.1), and Proposition 3.3,

Consequently, by the convex-hull extension of (1.7), discussed above, and the Cauchy–Schwarz inequality, we obtain

This implies almost sure exponential contraction of the diameter with a deterministic decay rate independent of the cutoffs. Unlike [GT20; DHYZ25], this argument does not use order preservation and is therefore applicable to the vector-valued model.

Bibliography