Scaling of the Sasamoto-Spohn model in equilibrium

We prove the convergence of the Sasamoto-Spohn model in equilibrium to the energy solution of the stochastic Burgers equation on the whole line. The proof, which relies on the second order Boltzmann-Gibbs principle, follows the approach of \cite{GJS} and does not use any spectral gap argument.


Model and results
The goal of this note is to show the convergence of a certain discretization of the stochastic Burgers equation: where W is a space-time white noise. This equation can be seen as the evolution of the slope of solutions to the KPZ equation [15] which is itself a model of an interface in a disordered environment. The KPZ/Burgers equation has been subject to an extensive body of work in the last years. It appears as the scaling limit of a wide range of particle systems [4,8], directed polymer models [3,20] and interacting diffusions [6], and constitutes a central element in a vast family of models known as the KPZ universality class [5,21].
Due to the nonlinearity, a lot of care has to be taken to obtain a notion of solution for (1). There are today several alternatives, for instance, regularity structure [14], paracontrolled distributions [11] and energy solutions [8,10,12], which is the approach we will follow.
While the discretization of the second derivative and noise are quite straightforward, there are a priori several ways to discretize the nonlinearity in Burgers equation. This particular choice is motivated by two reasons: first, it only involves nearest neighbor sites and, second, it yields the explicit invariant measure µ = ρ ⊗Z , where dρ(x) = 1 Our result states the convergence of the discrete equations (2) to Burgers equation in the sense of energy solutions (see Section 2 for a precise definition). Theorem 1. For each n ≥ 1, let u n be the solution to the system (2) for γ = n −1/4 and initial law µ, and let The sequence of processes (X n · ) n≥1 converges in distribution in C([0, T ], S ′ (R)) to the unique energy solution of the Burgers equation.
A similar result was shown in [11] for much more general initial conditions although restricted to the one-dimensional torus.
At the technical level, our approach relies on the techniques of [9] and avoids the use of any spectral gap estimate. The core of the proof consists in deriving certain dynamical estimates among which the so-called second order Boltzmann-Gibbs principle plays a major role. A key ingredient is a certain integration-by-parts satisfied by the model.
The paper is organized as follows: in Section 2, we recall the notion of energy solution from [8]. We show the invariance of the measure µ in Section 3. In Section 4, we prove the dynamical estimates. Finally, in Sections 5 and 6, we show, respectively, tightness and convergence to the energy solution. The construction of the dynamics (2) is given in the appendix.
We are now ready to formulate the definition of an energy solution: is a martingale with quadratic variation tE(∂ x ϕ), where A is the process from Theorem 2.
Existence of energy solutions was proved in [8]. Uniqueness was proved in [12].

Generator and invariant measure
The construction of the dynamics given by (2) is detailed in Appendix A. We denote by C the set of cylindrical functions F of the form F (u) = f (u −n , · · · , u n ), for some n ≥ 0, with f ∈ C 2 (R 2n+1 ) with polynomial growth of its partial derivatives up to order 2. The generator of the dynamics (2) acts on C as which formally correspond to the symmetric and anti-symmetric parts of L with respect We note that our model satisfies the Gaussian integration-by-parts formula: which will be heavily used in the sequel.
We will also consider the periodic model u M on Z M := Z/MZ and denote by L M , S M and A M the corresponding generator and its symmetric and anti-symmetric parts respectively. Finally, denote µ M = ρ ⊗Z M and let ρ M be its density.
Proof. The lemma follows from Echeverría's criterion ( [7], Thm 4.9.17) once we show with polynomial growth of its derivatives up to order 2. By standard integration-by-parts, It is a simple computation to show that S † M ρ M ≡ 0. It then remains to verify that But, using standard integration-by-parts once again, we can verify that there exists a degree three polynomial in two variables p(·, ·) such that Finally, Gaussian integration-by-parts yields a degree two polynomial in two variablesp(·, ·) such that which is telescopic. This ends the proof.
By construction of the infinite volume dynamics and taking the limit M → ∞, we obtain Corollary 1. The measure µ is invariant for the dynamics (2).

The second-order Boltzmann-Gibbs principle
We recall the Kipnis-Varadhan inequality: there exists C > 0 such that where the || · || −1,n -norm is defined through the variational formula The proof of this inequality in our context follows from a straightforward modification of the arguments of [12], Corollary 3.5. In our particular model, we have so that the variational formula becomes Denote by τ j the canonical shift τ j u i = u j+i and let − → u l j = 1 l l k=1 u j+k . Lemma 2. Let l ≥ 1 and let g be a function with zero mean with respect to µ which support does not intersect {1, · · · , l}. Let g j (s) = g(τ j u(s)). There exists a constant C > 0 such that Now, for f ∈ C , using integration-by-parts, by Young's inequality. Taking α = 2/n, we find that the above is bounded by which, thanks to the Kipnis-Varadhan inequality, shows that the left-hand-side of (4) is bounded by Finally, as g is centered, We now state the second-order Boltzmann-Gibbs principle: let Q(l, u) = ( − → u l 0 ) 2 − 1 l , Proposition 1. Let l ≥ 1. There exists a constant C > 0 such that Proof. We use the factorization We handle the first term with Lemma 2. The second term is treated in the following lemma.
Lemma 3. Let l ≥ 1. There exists a constant C > 0 such that For f ∈ C , using integration-by-parts, The second summand comes from the term i = 0. Hence,

By Young's inequality, this last expression is bounded by
Taking α = 2l/n, this is further bounded by The result then follows from the Kipnis-Varadhan inequality.

Tightness
In the sequel, we let ϕ ∈ S be a test function. Remember the fluctuation field is given by Recalling the definition of the operators S and A from Section 3, the symmetric and antisymmetric parts of the dynamics are given by where we used γ = n −1/4 . Then, the martingale part of the dynamics corresponds to and has quadratic variation We will use Mitoma's criterion [19]: a sequence Y n is tight in C([0, T ], S ′ (R)) if and only if Y n (ϕ) is tight in C([0, T ], R) for all ϕ ∈ S(R).

Martingale term.
We recall that M n (ϕ) = tE n (∇ n ϕ n ). From the Burkholder-Davis-Gundy inequality, it follows that E [|M n t (ϕ) − M n s (ϕ)| p ] ≤ C|t − s| p/2 E n (∇ n ϕ n ) p/2 , for all p ≥ 1. Tightness then follows from Kolmogorov criterion by taking p large enough.

Symmetric term. Tightness is obtained via a second moment computation and Kolmogorov criterion:
5.3. Anti-symmetric term. We study the tightness of the term We begin with a lemma: goes to zero in the ucp topology.
Proof. Using integration by parts, Using Young's inequality, by taking α = 2/n. Into the Kipnis-Varadhan inequality, this yields which shows that this process goes to zero in the ucp topology.
This means we can switch the term w j in the anti-symmetric part of the dynamics by u j u j+1 modulo a vanishing term. Note that, as we apply the previous lemma to a gradient, the constant term 1 will disappear. We are then left to prove the tightness of B n t (ϕ) = t 0 j u j (sn)u j+1 (sn)∇ n ϕ n j ds.
From Proposition 1, we have where, here and below, C denotes a constant which value can change from line to line. On the other hand, a careful L 2 computation, taking dependencies into account, shows that For t ≥ 1/n, we take l ∼ √ tn and get For t ≤ 1/n, a crude L 2 bound gives This gives tightness.

Convergence
From the previous section, we get processes X , S, B and M such that along a subsequence that we still denote by n. We will now identify these limiting processes.
6.1. Convergence at fixed times. A straightforward adaptation of the arguments in [6], Section 4.1.1, shows that X n t converges to a white noise for each fixed time t ∈ [0, T ]. This in turns proves that the limit satisfies property (S).
6.2. Martingale term. The quadratic variation of the martingale part satisfies By a criterion of Aldous [1], this implies convergence to the white noise.

Symmetric term. A second moment bound shows that
which shows that 6.4. Anti-symmetric term. We just have to identify the limit of the process B n (ϕ). Remembering the definition of the field X n , we observe that from where we get the convergences n, u(rn))∇ n ϕ n j dr.
The second limit follows by a suitable approximation of i ε (x) by S(R) functions (see [8], Section 5.3 for details). Now, by the second-order Boltzmann-Gibbs principle and stationarity, Taking l ∼ ε √ n and the limit as n → ∞ along the subsequence, The energy estimate (EC2) then follows by the triangle inequality. Theorem 2 yields the existence of the process Furthermore, from (5), we deduce that B = A.
It remains to check (EC1). It is enough to check that Using the smoothness of ϕ and a summation by parts, it is further enough to verify that For that purpose, we will use Kipnis-Varadhan inequality one last time: let f ∈ C , 2 n 1/4 j (u j+1 − u j )∇ n ϕ n j f dµ = 2 n 1/4 with α = 2/n, from where (6) follows.
Appendix A. Construction of the dynamics The system of equations (2) can be reformulated as where we used invariance in the last step. Next, we show tightness of the processes (in M) where we now identify u M with a periodic system on the line. This follows from Kolmogorov's criterion. It is enough to control expressions of type Together with a standard estimate on the increments of the Brownian motion, this yields E |u M j (t) − u M j (s)| 2 ≤ C|t − s| 2 . Hence, each coordinate is tight. By diagonalization, we can extract a subsequence of M k such that (u M k j ) converges in law in C[0, T ] for each j. This gives a meaning to the system (2).