Stationary Directed Polymers and Energy Solutions of the Burgers Equation

TL;DR

This paper demonstrates that the increments of the log-partition function in a semi-discrete directed polymer model converge to energy solutions of the stochastic Burgers equation, using a novel approach that bypasses traditional transforms and spectral gap estimates.

Contribution

It introduces a new proof technique for convergence to the stochastic Burgers equation without relying on the Cole-Hopf transform or spectral gap estimates.

Findings

Convergence of log-partition function increments to energy solutions of the stochastic Burgers equation.

Development of a second-order Boltzmann-Gibbs principle for the model.

A proof approach that simplifies analysis by avoiding spectral gap estimates.

Abstract

We consider the stationary O'Connell-Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.

Equations

$${\partial }_{t}h$$

$${\partial }_{t}u=\nu {\partial }_x^{2}u+\lambda {\partial }_x{u}^2+\sqrt{D}{\partial }_x\mathcal{W}.$$

$${\partial }_{t}Z=\nu {\partial }_x^{2}Z+\frac{\lambda \sqrt{D}}{\nu }Z\mathcal{W},$$

$$Z(t,x)={\mathbf{E}}_x\left[e^{{\int }_0^t\mathcal{W}(t-s,b_s)ds}\right],$$

$$Z_{\beta }(t,n)$$

$$\displaystyle\begin{aligned} Z_{\beta,\theta}(t,n)&=\hskip-20.0pt\int\limits_{-\infty<s_{0}<s_{1}<\dotsm<s_{n-1}<t}\hskip-30.0pt\exp\bigl{[}\beta B^{(0)}(s_{0})+\theta s_{0}+\beta\left\{B^{(1)}(s_{0},s_{1})+\dotsm+B^{(n)}(s_{n-1},t)\right\}\bigr{]}\,ds_{0,n-1},\end{aligned}$$

$$u_{\beta ,\theta }(t,j)=\log Z_{\beta ,\theta }(t,j)-\log Z_{\beta ,\theta }(t,j-1),j\ge 1.$$

$$e^{-u_{\beta ,\theta }(t,j)}\sim {\beta }^2\text{Gamma}({\beta }^{-2}\theta ).$$

$$d{u}_j=\left(e^{-u_j}-e^{-u_{j-1}}\right)dt+\beta \left(d{B}_t^{(j)}-d{B}_t^{(j-1)}\right)$$

$${\mathcal{X}}_t^n(\phi )=\sum _j(u^n(tn,j)-{\rho }_n)\phi (\frac{j-nt-a_n\sqrt{n}}{\sqrt{n}}).$$

$${\partial }_{t}u=\frac{1}{2}{\partial }_x^{2}u+c{\partial }_{x}u-\frac{1}{2}{\partial }_x{u}^2+{\partial }_x\mathcal{W},$$

$$d{u}_j=\left(V^{\prime }(u_{j-1})-V^{\prime }(u_j)\right)dt+\beta \left(d{B}_t^{(j)}-d{B}_t^{(j-1)}\right),$$

$${\phi }_j^n$$

$${\nabla }^n{\phi }_j^n$$

$${\Delta }^n{\phi }_j^n$$

$${\mathcal{E}}_n(\phi )=\frac{1}{\sqrt{n}}\sum _j{\phi }_j^2,\mathcal{E}(\phi )=\int {\phi }^2(x)dx.$$

$${\partial }_{t}u=\frac{1}{2}{\partial }_x^{2}u+c{\partial }_{x}u-\frac{1}{2}{\partial }_x{u}^2+{\partial }_x\mathcal{W},$$

$$\mathbb{E}\left[\mathcal{W}(f_1)\mathcal{W}(f_2)\right]={\int }_{{\mathbb{R}}_{+}\times \mathbb{R}}{f}_1(t,x)f_2(t,x)dxdt,f_1,f_2\in L^2({\mathbb{R}}_{+}\times \mathbb{R}).$$

$${\mathcal{A}}_{s,t}^{\epsilon }(\phi )={\int }_s^t{\int }_{\mathbb{R}}{u}_r(i_{\epsilon }(x){)}^2{\partial }_x\phi (x)dxdr$$

$$\mathbb{E}\left[{\left|{\int }_s^t{u}_r({\partial }_x^2\phi )dr\right|}^2\right]\le \kappa (t-s)\mathcal{E}({\partial }_x\phi ),$$

$$\mathbb{E}\left[{\left|{\mathcal{A}}_{s,t}^{\epsilon }(\phi )-{\mathcal{A}}_{s,t}^{\delta }(\phi )\right|}^2\right]\le \kappa (t-s)\epsilon \mathcal{E}({\partial }_x\phi ).$$

$${\mathcal{A}}_t(\phi )=\underset{\epsilon \to 0}{\lim }{\mathcal{A}}_{0,t}^{\epsilon }(\phi ),$$

$$u_t(\phi )-u_0(\phi )-\frac{1}{2}{\int }_0^t{u}_s({\partial }_x^2\phi )ds-c{\int }_0^t{u}_s({\partial }_x\phi )ds-{\mathcal{A}}_t(\phi )$$

$$u_{T-t}(\phi )-u_T(\phi )-\frac{1}{2}{\int }_0^t{u}_{T-s}({\partial }_x^2\phi )ds+c{\int }_0^t{u}_{T-s}({\partial }_x\phi )ds+{\mathcal{A}}_T(\phi )-{\mathcal{A}}_{T-t}(\phi )$$

$$d{u}_j$$

$$d{u}_1$$

$$d{u}_j$$

$$u_j(0)=-\log X_j+\frac{1}{2}\log n,j\ge 1,$$

$$L$$

$${\mathcal{X}}_t^n(\phi )=\sum _j(u_j(tn)-{\rho }_n){\phi }_j^n,$$

$${\mathcal{M}}_t^n(\phi )={\mathcal{X}}_t^n(\phi )-{\mathcal{X}}_0^n(\phi )-{\int }_0^t\left({\partial }_s+nL\right){\mathcal{X}}_s^n(\phi )ds=\beta {\int }_0^t\sum _j{\nabla }^n{\phi }_j^{n}d{B}_s^{(j)}$$

$$⟨{\mathcal{M}}^n(\phi ){⟩}_t=\frac{{\beta }^2}{2}{\int }_0^t\sum _j({\nabla }^n{\phi }_j^n{)}^{2}ds.$$

Full text

The proof does not rely on the Cole-Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann-Gibbs principle.

AMS 2010 subject classifications. Primary 60H15 secondary 60K35, 82B44

Key words and phrases. KPZ Equation, Burgers Equation, Directed Polymers

1 Instituto de Matemática Pura e Aplicada. Partially supported by CNPq and FAPERJ

2 Pontificia Universidad Católica de Chile. Partially supported by Fondecyt grant 1171257 and Núcleo Milenio ‘Modelos Estocásticos de Sistemas Complejos y Desordenados’

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovative programme (grant agreement No 715734), and from MATH Amsud ‘Random Structures and Processes in Statistical Mechanics’

1. Introduction, Model and Results

1.1. KPZ equation and Stochastic Burgers equation

The Kardar-Parisi-Zhang equation [33], or KPZ equation, was introduced in the physics literature as a model for interface motions in generic situations. The typical physical set-up is the following: suppose we have a thin physical system where a stable and a meta-stable phase can coexist and suppose both phases are separated by an interface. We are concerned with the behaviour of such an interface as the stable phase invades the meta-stable region. The first thing one observes is a net motion of the interface, meaning that it has in average a non-zero velocity. At a closer look, we can observe very intricate fluctuations with an atypical order of magnitude of $t^{1/3}$ and highly non-Gaussian statistics. Assuming that the position of the interface can be (locally) described by a height function $h(t,x)$, the authors of [33] conclude that its dynamics is governed by the equation

[TABLE]

where $\mathscr{W}$ is a space-time white noise and $\nu$, $\lambda$ and $D$ are constants that depend on the precise model under consideration. In particular, the quantity $\sqrt{D}$ represents the intensity of the noise which, in the terminology of [45], represents the random back and forth between the two phases.

Perhaps the most accurate experimental realization of this dynamics is given by the growing interfaces in liquid crystal turbulence (see [46] and references therein). There, a thin film of turbulent liquid crystal is kept out of thermal equilibrium. Then, a seed of the stable phase is created and grows as a cluster. The statistics of the fluctuation of the interface separating the two phases match the theoretical predictions with spectacular accuracy.

The KPZ equation constitutes a particular representative of a huge family of models known as the KPZ universality class. These models are characterized by displaying cube-root fluctuations which laws rescale to non-Gaussian distributions that first appeared in random matrix theory. Despite their complicated nature, some of these models have even explicit laws that allow for fine asymptotic analysis. The study of these models has generated an huge body of work in the mathematics and physics communities that is impossible to summarize in a concise way. We refer the reader to the reviews [15], [12] and [47] for a recent exposition of the state of the art.

Here, we will not be concerned with the ‘integrable’ nature of the KPZ universality class but will instead focus on a particular model which, in a very precise regime, rescales to the KPZ equation. The emergence of KPZ as scaling of discrete models first appeared in the work [7] for the weakly asymmetric exclusion process, the main representative of the so-called weakly asymmetric limits. This type of limit has since then appeared in many contexts, for instance [5, 17, 18, 32] among others. Our setting is an example of a different kind of limit, the intermediate disorder regime, first observed in [2] (see also [38, 13]).

The KPZ equation has two ‘avatars’: the stochastic Burgers equation and the stochastic heat equation. Letting $u=\partial_{x}h$ be the slope of $h$, we can see that $u$ satisfies the equation

[TABLE]

This is known as the stochastic Burgers equation. The notion of solutions for the KPZ and Burgers equation is mathematically very delicate. In the KPZ equation, the best space regularity one can expect is that of a Brownian motion. As such, its first derivative is a genuine distribution and its square requires a very careful treatment to be properly defined. Burgers equation of course shares a similar problems and is in fact distribution-valued.

An early solution to this issue consisted in taking a clever non-linear transform of $h$ which removes the non-linear part of the equation: let $\displaystyle Z(t,x)=\exp\left\{\frac{\lambda}{\nu}\log h(t,x)\right\}$. Then, $Z$ satisfies the equation

[TABLE]

known as the stochastic heat equation (SHE). This equation can be solved by ad-hoc methods and, then, the solution to KPZ can be ‘defined’ as $\displaystyle h(t,x)=\frac{\nu}{\lambda}\log Z(t,x)$. This is the so-called Cole-Hopf solution to the KPZ equation which can be traced back at least to [7]. Although this provides a notion of solution which is useful for many purposes (such as showing the convergence of a wide family of discrete models), it is unsatisfactory in the sense that it does not show that $h$ actually satisfies an equation. We will come back to the Cole-Hopf solution in the next section as it provides a natural link between KPZ/Burgers/SHE and directed polymers.

In recent years, more robust theories of existence and uniqueness of KPZ/Burgers equation have emerged. The first one [26] was pioneered by M. Hairer and lead to the development of the theory of regularity structures [27]. This theory allows to give a solid notion of solution for a wide family of singular stochastic PDEs as well as providing a framework to prove the convergence of discrete models [28, 8].

This breakthrough was shortly followed by an existence and uniqueness theory for KPZ/Burgers in the framework of paracontrolled distributions [22]. As in the case of regularity structures, this theory can be used to treat other stochastic PDEs beyond KPZ [9] and is amenable to show the convergence of discrete models [10, 35]. Furthermore, this theory can be succesfully applied to the KPZ equation on the whole real line [41].

A third approach is provided by the theory of energy solutions introduced in [17] and further developed in [20]. In this approach, Burgers equation is formulated as a martingale problem. Uniqueness for this weak formulation on the whole line was proved in [23]. Since [17], this approach was successfully applied to show the convergence of many discrete models to the KPZ/ Burgers equation [18, 19, 14, 32]. One substantial advantage is that it requires very weak quantitative estimates. So far, this theory is mainly restricted to the stationary setting (see however [24, 25]). Let us go back to (1). If $\lambda=0$, the equation becomes a linear stochastic heat equation with additive noise. Its solutions are explicit, Gaussian and have the spatial white noise as an invariant measure. One remarkable fact is that this invariant measure is preserved by the addition of the non-linear term. In the KPZ setting, the white noise initial condition corresponds to a double-sided Brownian motion. Of course, even if at fixed times the spatial distribution of the process is fairly simple, the time-correlations are extremely complicated. This will be the context considered in this work.

1.2. The Cole-Hopf transformation, the Stochastic Heat Equation and Directed Polymer

The Cole-Hopf transform provided a way to give a meaning to the solutions of the KPZ equation in terms of the solutions of the SHE. For the SHE, solutions can be described by means of a chaos expansion, taking advantage of the particular structure of the equation [47]. This also provided a starting point to show the convergence of discrete models which themselves satisfy a discrete version of the Cole-Hopf transform, for instance the exclusion process [7, 5] (for which the discrete Cole-Hopf transform dates back to [16]), directed polymers [2, 38] which are naturally formulated at the level of the SHE, one-dimensional random walks in vanishing random environments [13] and weakly asymmetric bridges [34] among others. Once again, we point the reader to the reviews [12] and [47] for a more detailed summary of the field.

This approach has two limitations. First, as it actually proves convergence to the SHE, it does not in principle provide convergence to KPZ/Burgers (regardless of the way these are interpreted). For instance, it does not allow to obtain direct convergence to Burgers equation for the occupation field of the exclusion process as in [17] although it can be used to show convergence of its height function to the Cole-Hopf solution of KPZ. Second, this approach relies heavily on the availability of a discrete Cole-Hopf transform which is not available for relevant models such as the Sasamoto-Spohn model [42] or the coupled diffusions considered in [14].

These difficulties can be circumvented using the theories described in the previous section. The works [28, 8] provide a general framework to treat discrete models in the context of regularity structures. In the framework of paracontrolled distributions, the work [22] was successful in giving a first proof of the convergence of the periodic Sasamoto-Spohn model and the works [10, 35] developed robust arguments to treat stochastic PDEs on the lattice. In the context of energy solutions, speed-change exclusion dynamics was treated in [17, 18], interacting diffusions in [14] and the Sasamoto-Spohn model on the whole line in [32]. In all these last examples, direct convergence to Burgers equation for the fluctuation field is proved.

One interesting fact about the Cole-Hopf solution is that it gives a direct link between directed polymers and the KPZ/SHE equation. Consider the SHE (2) with initial condition $Z(0,\cdot)=\delta_{0}$ and assume for a moment that the potential $\mathscr{W}$ is smooth. Then, Feynman-Kac formula yields the explicit solution (with $D=1$ and $\lambda=\nu=\frac{1}{2}$)

[TABLE]

where $\mathbf{E}_{x}$ is the expectation with respect to the law of a Brownian bridge $(b_{s}:\,0\leq s\leq t)$ with $b_{0}=0$ and $b_{t}=x$. As such, $Z$ can be viewed as the partition function of a directed polymer model where the energy of a path is given by $H(b)=\int^{t}_{0}\mathscr{W}(s,b_{s})\,ds$. These directed polymers in random environment were introduced in the physics literature as a model for the roughening of interfaces in random environment [31]. They have been the object of a vast body of mathematical work since then (see [11] for a recent monograph on the subject). When $\mathscr{W}$ is taken as a white noise, it is possible to give a sense to (3) (see [47]) and even to a continuum polymer measure [3].

1.3. Semi-discrete Directed Polymers in a Brownian Environment and the Main Result

Polymer models are defined by specifying a path space and an environment. We will work exclusively with the model of Directed Polymers in a Brownian Environment introduced in [40]. In this case:

• •

Polymer paths are nondecreasing càdlàg paths $x:[0,t]\to\mathbb{N}$ with nearest-neighbor jumps, $x(0)=1$, and $x(t)=n$. A path can be coded in terms of its jump times $0=s_{0}<s_{1}<\cdots<s_{n-1}<s_{n}=t$.

• •

The environment consists of a family of independent double-sided one-dimensional standard Brownian motions $\{B^{(j)}(\cdot):j\geq 1\}$ with $B^{(j)}(0)=0$.

At level $j$, the path collects the increment $B^{(j)}(s_{j-1},s_{j})=B^{(j)}(s_{j})-B^{(j)}(s_{j-1})$. The partition function in a fixed Brownian environment is defined as

[TABLE]

for $(n,t)\in\mathbb{N}\times[0,\infty)$, where $ds_{1,n-1}$ is a short-hand notation for $ds_{1}\dotsm ds_{n-1}$.

We will be mainly concerned with a stationary version of the model: enlarge the environment by adding another Brownian motion $B^{(0)}$ independent of $\{B^{(j)}:\,j\geq 1\}$ and introduce a new parameter $\theta\in(0,\infty)$. The stationary partition function is defined as

[TABLE]

for $n\geq 1$ and $Z_{\beta,\theta}(t,0)=\exp\{\beta B^{(0)}(t)+\theta t\}$. Here, stationarity refers to a specific property of the model highlighted in [40]: define

[TABLE]

Then, for all $t\geq 0$, $\{u_{\beta,\theta}(t,j):\,j\geq 1\}$ is an i.i.d. family with

[TABLE]

In other words, if we denote by $m_{\beta,\theta}$ the law of $-\log X-\log\beta^{2}$ where $X\sim\text{Gamma}(\beta^{-2}\theta)$ and $\mu_{\beta,\theta}=m_{\beta,\theta}^{\otimes\mathbb{N}}$, then $\mu_{\beta,\theta}$ is the stationary measure of the process $\{u_{\beta,\theta}(\cdot,j):\,j\geq 1\}$.

The processes $Z_{\beta,\theta}$, $h_{\beta,\theta}=\log Z_{\beta,\theta}$ and $u_{\beta,\theta}$ can be seen as semi-discrete stationary versions of the stationary SHE, KPZ and Burgers equations respectively. The link between $Z_{\beta,\theta}$ and the SHE can be seen as a rigorous version of the Feynman-Kac formula (3). It turns out that these processes actually satisfy lattice versions of these equations. In the case of Burgers, an application of Itô’s formula shows that $u_{j}(\cdot)=u_{\beta,\theta}(\cdot,j)$ satisfies the system of stochastic differential equations

[TABLE]

A naive Taylor expansion suggests that $e^{-u_{j}}-e^{-u_{j-1}}\simeq(u_{j-1}-u_{j})+\frac{1}{2}(u_{j}^{2}-u_{j-1}^{2})$. In an appropriate skew scaling, the discrete gradient $u_{j-1}-u_{j}$ becomes a second derivative while the difference of squares is reminiscent of the term $\partial_{x}u^{2}$ in (1). We will show that this is indeed the case although it does not follow from such a simple argument. It is actually the core of our proof. The system above was already observed in [44]. Note that complicated non-linearities leading to Burgers equation where considered in the works [29, 30] and later in [21] in the context of energy solutions. The works [30] and [21] deal with general examples of weakly asymmetric scaling. Our work is an example of intermediate disorder scaling. The reference [29] deals with both settings. One advantage of the energy solution approach is that it allows to consider models on the whole real line although only in equilibrium.

We can state our result: denote $u_{\beta,\theta}$ by $u^{n}$ whenever $\beta=n^{-1/4}$ and $\theta=1+\frac{1}{2\sqrt{n}}$ with stationary initial condition. We fix once for all an increasing diverging sequence $(a_{n})_{n}$ such that $\displaystyle\lim_{n\to\infty}\frac{a_{n}}{\sqrt{n}}=0$. Let $\rho_{n}:=\mathbb{E}[u^{n}]$ and define the fluctuation field $\mathcal{X}^{n}$ acting on test functions by

[TABLE]

Theorem 1**.**

The sequence of processes $(\mathcal{X}^{n}_{t}:\,t\in[0,T])_{n\geq 1}$ converges in distribution in $C([0,T],\mathcal{S}^{\prime}(\mathbb{R}))$ to the unique energy solution of the Burgers equation

[TABLE]

where $c\in\mathbb{R}$ is an explicit constant and $\mathscr{W}$ is a space-time white noise.

Remark 1**.**

It is reasonable to expect a non-trivial transport term as there are several sources of asymmetry in the model. First, our scaling was meant to properly normalize the partition function. As such, its logarithm is slightly off-center. Second, the direction $(1,1)$ is not exactly the characteristic direction or, equivalently, to force it to be characteristic, a more careful choice of constants has to be made (see [43, 39]). Of course, both settings are asymptotically equivalent. Finally, this model can be seen as a system of coupled diffusions in the highly non-symmetric potential $V(u)=e^{-u}-1+u$. This is another source of asymmetry. The precise value of $c$ can be obtained by careful book-keeping along the proof. We found it to be $-\frac{9}{10}$. In any case, this transport term can be removed by a change of coordinates in the equation and, at the discrete level, with a more careful centering of the test functions.

Remark 2**.**

The sequence $(a_{n})_{n}$ is introduced to deal with the fact that the discrete model is defined only for $j\geq 1$. For any compactly supported test function, the fluctuation field will then be well defined for $n$ large enough. As the system is stationary, this correction is harmless.

Remark 3**.**

We note that the result above can be easily generalized to systems of SDEs of the type

[TABLE]

where $V$ is a real-valued function which is quadratic at [math] and has appropriate growth at $\pm\infty$, provided the dynamics above can be properly defined. The existence of the dynamics is a difficult question (see [36] for results in this direction). In our case (where $V(u)=e^{-u}-1+u$), the interpretation of the system (4) in terms of directed polymers settles the issue but, in general, this connection is lost. We will not consider such a general framework in this article.

1.4. Structure of the Article

In Section 2, we recall the notion of energy solutions of Burgers equation. In Section 3, we carefully state the system of SDEs satisfied by the model, identify its different components and give a martingale interpretation. In Section 4, we present some useful estimates on the moments of the discrete model. In Section 5, we prove the dynamical estimates which are the core of our proof. In particular, we prove the second order Boltzmann-Gibbs principle in Section 5.2. In Section 6, we prove the tightness of the fluctuation field and identify its limit in Section 7. The Appendix contains additional estimates needed in Sections 6 and 7.

1.5. General Notations

Recall that we fixed an increasing diverging sequence $(a_{n})_{n}$ such that $\displaystyle\lim_{n\to\infty}\frac{a_{n}}{\sqrt{n}}=0$. For test functions $\varphi$, we define

[TABLE]

Note that, even though the discretization depends on the value of $t$, we remove it from the notation as no confusion will arise. For sequences $(\varphi_{j})_{j}$ (resp. test functions $\varphi$), we define

[TABLE]

We denote the law of the stationary process $u^{n}$ by $\mathbb{P}_{n}$ and expected value with respect to $\mathbb{P}_{n}$ by $\mathbb{E}_{n}$. As such, $u^{n}$ will be simply denoted by $u$, which can be seen as the canonical process under the law $\mathbb{P}_{n}$. Note that, in this context, $\beta^{-2}\theta=\sqrt{n}+1/2$. We denote by $m_{n}$ the law of $-\log X+\tfrac{1}{2}\log n$ where $X\sim\text{Gamma}(\sqrt{n}+1/2)$ and $\mu_{n}=m_{n}^{\otimes\mathbb{N}}$ which, according to the previous discussion, turns out to be the stationary measure for $u$. As usual, $C$ denotes a positive constant whose value may change from line to line.

2. Energy solutions of the Burgers equation

We will present the basics of the theory of energy solutions of the stochastic Burgers equation as it was introduced in [17] and further developed in [20, 23] (see also [19, 24, 25]). Recall we are concerned with the equation

[TABLE]

where $\mathscr{W}$ is a space-time white noise, i.e. a distribution-valued centered Gaussian process with covariance $\mathbb{E}[\mathscr{W}(t,x)\mathscr{W}(s,y)]=\delta(t-s)\delta(y-x)$. More precisely, $\mathscr{W}$ acts on $L^{2}(\mathbb{R}_{+}\times\mathbb{R})$ in such a way that the random variables $\{\mathscr{W}(f):\,f\in L^{2}(\mathbb{R}_{+}\times\mathbb{R})\}$ are jointly Gaussian with covariance

[TABLE]

Due to the singularity of the noise, solutions to (5) can only be expected to be distribution-valued in space. The main difficulty then consists in giving a consistent meaning to the term $\partial_{x}u^{2}$. As we will see below, it is possible to make sense to this expression as a space-time distribution.

We start with a definition:

Definition 1**.**

We say that a process $\{u_{t}:\,t\in[0,T]\}$ satisfies condition (S) if, for all $t\in[0,T]$, the $\mathcal{S}^{\prime}(\mathbb{R})$-valued random variable $u_{t}$ is a white noise of variance $1$.

For a process $\{u_{t}:\,t\in[0,T]\}$ satisfying condition (S), $0\leq s<t\leq T$, $\varphi\in\mathcal{S}(\mathbb{R})$ and $\varepsilon>0$, we define

[TABLE]

where $i_{\varepsilon}(x)=\varepsilon^{-1}{\bf 1}_{(x,x+\varepsilon]}$.

Definition 2**.**

Let $\{u_{t}:\,t\in[0,T]\}$ be a process satisfying condition (S). We say that $\{u_{t}:\,t\in[0,T]\}$ satisfies the energy estimate if there exists a constant $\kappa>0$ such that:

(EC1) For any $\varphi\in\mathcal{S}(\mathbb{R})$ and any $0\leq s<t\leq T$,

[TABLE]

(EC2) For any $\varphi\in\mathcal{S}(\mathbb{R})$, any $0\leq s<t\leq T$ and any $0<\delta<\varepsilon<1$,

[TABLE]

We state a key theorem proved in [17] which allows to give a sense to the quadratic term in (5):

Theorem 2**.**

Assume $\{u_{t}:\,t\in[0,T]\}$ satisfies (S) and (EC2). Then, there exists an $\mathcal{S}^{\prime}(\mathbb{R})$-valued stochastic process $\{\mathcal{A}_{t}:\,t\in[0,T]\}$ with continuous paths such that

[TABLE]

in $L^{2}$, for any $t\in[0,T]$ and $\varphi\in\mathcal{S}(\mathbb{R})$.

We are now ready to formulate the definition of an energy solution:

Definition 3**.**

We say that $\{u_{t}:\,t\in[0,T]\}$ is a stationary energy solution of the stochastic Burgers equation (5) if

1. 1.-

$\{u_{t}:\,t\in[0,T]\}$* satisfies (S), (EC1) and (EC2).* 2. 2.-

For all $\varphi\in\mathcal{S}(\mathbb{R})$, the process

[TABLE]

is a martingale with quadratic variation $t\mathcal{E}(\partial_{x}\varphi)$, where $\mathcal{A}$ is the process from Theorem 2. 3. 3.-

For all $\varphi\in\mathcal{S}(\mathbb{R})$, the process

[TABLE]

is a martingale with quadratic variation $t\mathcal{E}(\partial_{x}\varphi)$.

Existence of energy solutions was proved in [17]. Uniqueness was proved in [23].

3. System of SDEs and the Martingale Decomposition

An application of Itô’s formula shows that, under $\mathbb{P}_{n}$, the collection $\{u_{j}:\,j\geq 1\}$ satisfies the system of SDEs:

[TABLE]

where $W_{j}=1-e^{-u_{j}}$ and $\beta=n^{-1/4}$. As it will be noticed later, $\mathbb{E}_{n}[W_{j}]=-\beta^{2}/2$. Writing $\overline{W}_{j}=W_{j}-\mathbb{E}_{n}[W_{j}]$ and setting $\overline{W}_{0}=0$, the system above can be summarized as

[TABLE]

The initial condition is taken as

[TABLE]

where $(X_{j})_{j}$ is an i.i.d. family of $\text{Gamma}(\sqrt{n}+1/2)$ random variables. Hence, the generator of this dynamics acts on smooth cylindrical functions as

[TABLE]

where $\partial_{j}=\partial/\partial_{u_{j}}$. Remembering the definition of the density field

[TABLE]

Dynkin’s formula implies that

[TABLE]

is a martingale with quadratic variation

[TABLE]

Note that the time integral cannot be removed as the discretization of $\varphi$ depends on time. By integration-by-parts, we can formally obtain $L^{*}$, the adjoint of $L$ in $L^{2}(\mu_{n})$:

[TABLE]

This allows us to identify the symmetric and anti-symmetric parts of the generator:

[TABLE]

With this at hands, we can properly decompose the dynamics: remembering $\beta=n^{-1/4}$, the symmetric part corresponds to

[TABLE]

while the anti-symmetric part corresponds to

[TABLE]

4. Static Estimates

We briefly recall some facts about the Gamma and log-Gamma distributions. If $X\sim\text{Gamma}(\nu)$, then

[TABLE]

where $\Gamma(\nu)=\int^{\infty}_{0}y^{\nu-1}e^{-y}dy$ is the Gamma function. By explicit computations,

[TABLE]

Now, if we take $\beta=n^{-1/4}$, $\theta=1+1/(2\sqrt{n})$ and let $\nu=\beta^{-2}\theta=\sqrt{n}+1/2$, we obtain

[TABLE]

as, under $\mathbb{P}_{n}$, $e^{-u}\sim\beta^{2}X$ with $X\sim\text{Gamma}(\sqrt{n}+1/2)$. Here, ${\rm\mathbb{V}ar}_{n}$ denotes the variance with respect to $\mathbb{P}_{n}$. On the other hand, for $X\sim\text{Gamma}(\sqrt{n}+1/2)$,

[TABLE]

from where we can compute

[TABLE]

with $\Psi_{0}=\Gamma^{\prime}/\Gamma$ and $\Psi_{1}=\Psi_{0}^{\prime}$. Asymptotics of these functions are known [1]:

[TABLE]

From this, we conclude that

[TABLE]

The following lemma provides bounds for higher moments:

Lemma 1**.**

Let $F$ be a locally bounded function such that $|F(x)|e^{-c|x|}$ is bounded for some constant $c>0$ and such that there exists $C>0$, $a>1$ and $k\geq 1$ such that

[TABLE]

Then, there exists $C^{\prime}>0$ such that

[TABLE]

Proof.

Write again $\beta=n^{-1/4}$, $\theta=1+1/(2\sqrt{n})$ and $\nu=\beta^{-2}\theta=\sqrt{n}+1/2$. First, an application of Stirling’s formula shows that

[TABLE]

for some $c>0$. Next, allowing the value of $C$ to change from line to line,

[TABLE]

for some $c_{1}>0$. We bound the contributions of $[-a,a]^{c}$: for some small enough $c_{2}>0$, we have

[TABLE]

and

[TABLE]

for some $c_{3}>0$. ∎

In particular, for each $k\geq 1$, we can find constants $C_{k}>0$ such that

[TABLE]

5. Dynamical estimates

We denote by $\mathscr{C}$ the collection of cylindrical functions $F$ of the form $F(u)=f(u_{-n},\cdots,u_{n})$ for some $n\geq 0$ and some $f\in C^{2}(\mathbb{R}^{2n+1})$ with polynomial growth of its derivatives up to order $2$. We recall the Kipnis-Varadhan estimate:

[TABLE]

where the $||\cdot||_{-1,n}$-norm is defined through the variational formula

[TABLE]

The proof is a straightforward adaptation of [14], Corollary 3.5. Note that

[TABLE]

so that

[TABLE]

Next, we notice that our model satisfies the integration-by-parts formula:

[TABLE]

5.1. One-block estimate

Recall $\overline{W}_{j}=W_{j}-\mathbb{E}_{n}[W_{j}]$ and let $\overrightarrow{W}^{l}_{j}=\displaystyle\frac{1}{l}\sum^{j+l-1}_{k=j}\overline{W}_{k}$ for $l\geq 2$. Let also $\tau_{j}$ denote the canonical shift: $\tau_{j}u_{i}=u_{i+j}$. In the following, we consider test functions $(\varphi_{j})_{j}$ which may depend on time.

Lemma 2**.**

Let $l\geq 2$ and let $g$ be a function with zero-mean respect to $\mu_{n}$ such that $g(u)=\tilde{g}(u_{j_{0}})$ for some $\tilde{g}:\mathbb{R}\to\mathbb{R}$ and $j_{0}\notin\{0,\cdots,l-1\}$. Write $g_{j}=g(\tau_{j}u)$. There exists a constant $C>0$ such that

[TABLE]

where $\mathcal{E}_{n}(\varphi)=\displaystyle\frac{1}{\sqrt{n}}\sum_{j}\varphi_{j}^{2}$.

Proof.

First, we observe that

[TABLE]

for $\psi_{i}=(l-i)/l$. Rearranging the sum (simply put $k=j+i$),

[TABLE]

where $F_{k}:=\sum^{l-2}_{i=0}\varphi_{k-i}g_{k-i}\psi_{i}$. Hence, for $f\in\mathscr{C}$, using integration-by-parts and our hypothesis on $g$,

[TABLE]

by Young’s inequality and $\beta^{2}\sqrt{n}=1$. Taking $\alpha=2n^{-1/2}$, we get that the above is bounded by

[TABLE]

which yields the bound

[TABLE]

Finally,

[TABLE]

The result follows from the Kipnis-Varhadan estimate. ∎

5.2. The second-order Boltzmann-Gibbs principle

Let

[TABLE]

The following is the central estimate in our proof:

Proposition 1**.**

[TABLE]

Proof.

Decompose as follows:

[TABLE]

The first term is handled with the one-block estimate with $g=\overline{W}_{-1}$ together with $\mathbb{E}_{n}[|\overline{W}|^{2}]\leq Cn^{-1/2}$ and gives the bound with the $l/\sqrt{n}$ term. The second one is the object of the next lemma and gives the same bound. The third one can be estimated by a careful $L^{2}$ computation and gives the bound with the $T/l^{2}$ term: using $\beta^{2}\sqrt{n}=1$, applying Jensen’s inequality, Tonelli and stationarity,

[TABLE]

Next, we have to take dependencies into account to compute the expected value: using again Jensen’s inequality and the independence of $\overrightarrow{W}^{l}_{j}$ and $\overrightarrow{W}^{l}_{k}$ if $|j-k|\geq l$,

[TABLE]

as $\mathbb{E}_{n}[|\overrightarrow{W}^{l}|^{2}]\leq C/l\sqrt{n}$. ∎

The following lemma finishes the proof of the Boltzmann-Gibbs principle:

Lemma 3**.**

[TABLE]

Proof.

Let $f\in\mathscr{C}$. We begin with a computation:

[TABLE]

where $\psi_{k}=(l-k)/l$. We will apply integration-by-parts: for $k\geq 1$,

[TABLE]

The term $k=0$ has to be handled separately:

[TABLE]

Carefully recombining the terms yields the identity

[TABLE]

By Young’s inequality, twice the above is bounded by

[TABLE]

Taking $\alpha=2l/\sqrt{n}$ and using $\beta^{2}\sqrt{n}=1$, the bound becomes

[TABLE]

The result follows from Kipnis-Varadhan inequality and the bound $\displaystyle\mathbb{E}_{n}[(\overrightarrow{W}^{l}_{j})^{2}]\leq C\frac{1}{l\sqrt{n}}$. ∎

6. Tightness

We will use Mitoma’s criterion [37]: a sequence $(\mathcal{Y}^{n})_{n}$ is tight in $C([0,T],\mathcal{S}^{\prime}(\mathbb{R}))$ if and only if $(\mathcal{Y}^{n}(\varphi))_{n}$ is tight in $C([0,T],\mathbb{R})$ for all $\varphi\in\mathcal{S}(\mathbb{R})$.

6.1. Martingale term

Recall that

[TABLE]

has quadratic variation

[TABLE]

Hence, from the Burkholder-Davis-Gundy inequality, it follows that

[TABLE]

for all $p\geq 1$. Tightness follows from Kolmogorov’s criterion by taking $p$ large enough.

6.2. Symmetric term

Recall that

[TABLE]

Tightness follows at once from an $L^{2}$ bound:

[TABLE]

where we used $\mathbb{E}_{n}\left[\overline{W}(sn)^{2}\right]=O(n^{-1/2})$.

6.3. Anti-symmetric term

Recall

[TABLE]

where

[TABLE]

Using $\mathbb{E}_{n}[|u_{j}(sn)-\rho_{n}|^{2}]=O(n^{-1/2})$ and the mean-value theorem,

[TABLE]

As a consequence, $E^{n}_{\cdot}(\varphi)$ converges to [math] in the ucp topology.

Now, a naive Taylor expansion suggests that

[TABLE]

explaining in particular the emergence of the quadratic term. This simple argument has two flaws: first, we are unable to handle the quadratic term as is, and second, the order three terms cannot be neglected based on moments considerations only. However, order four and higher terms can be neglected:

[TABLE]

A similar bound holds for powers of $\overline{W}$. We proceed now to a Taylor expansion which will be more useful to us: first,

[TABLE]

Here, the error of order $u^{4}_{j}$ takes into consideration positive values of $u_{j}$ while the term $|W_{j}|^{5}$ is included to account for large negative values of $u_{j}$. On the other hand,

[TABLE]

Equating both expansions and setting $a=-1/2$ and $b=-1/12$, we obtain

[TABLE]

Keeping in mind the nature of our dynamical estimates, we must find a way to ‘shift’ the index of one of the terms in each product in the right-hand-side. We use the identities

[TABLE]

yielding

[TABLE]

We use Taylor expansions one last time to switch between $W_{j-1}u_{j}$ and $W_{j-1}W_{j}$:

[TABLE]

Hence,

[TABLE]

We will investigate the convergence of each of these terms separately. The first (and main) term will be treated at the end of the section. The analysis of the second term is rather lengthy and will be left for the appendix. The terms involving $L$ are treated in Lemma 4 and 5 below. The term $\beta^{2}u_{j}$ is easily seen to be tight. Finally, the term involving $W_{j-1}u^{3}_{j}$ can be neglected by means of an $L^{2}$ computation.

Lemma 4**.**

There exists a constant $C>0$ such that

[TABLE]

Proof.

Let $g\in\mathscr{C}$. By integration-by-parts,

[TABLE]

Hence, by Young’s inequality,

[TABLE]

With $\alpha=8/\sqrt{n}$, this is further bounded by

[TABLE]

The result follows from Kipnis-Varadhan inequality. ∎

Lemma 5**.**

There exists a constant $C>0$ such that

[TABLE]

Proof.

Let $g\in\mathscr{C}$. By integration-by-parts,

[TABLE]

The proof is then similar to the previous lemma. ∎

We now focus on the term $W_{j-1}W_{j}$. Note that

[TABLE]

An $L^{2}$ computation easily shows that the contribution of the linear terms is tight. The term $\frac{1}{n}$ will disappear as we only test against gradients. We are left to show the tightness of the term

[TABLE]

By Proposition 1 and stationarity,

[TABLE]

On the other hand, a careful $L^{2}$ computation taking dependencies into account shows that

[TABLE]

For $\frac{1}{n}\leq t_{2}-t_{1}\leq 1$, we can take $l\sim\sqrt{(t_{2}-t_{1})n}$ in the above two inequalities to get

[TABLE]

For $t_{2}-t_{1}\leq\frac{1}{n}$, a crude $L^{2}$ bound yields

[TABLE]

This proves tightness.

7. Identification of the Limit

By tightness, we obtain processes $\mathcal{X},\,\mathcal{S},\,\widetilde{\mathcal{B}}$ and $\mathcal{M}$ such that

[TABLE]

along a subsequence that we still denote by $n$.

7.1. Convergence at fixed times

A straightforward adaptation of the arguments in [14], Section 4.1.1, shows that $\mathcal{X}^{n}_{t}$ converges to a white noise for each fixed time $t\in[0,T]$. This in turns proves that the limit satisfies property (S).

7.2. Linear terms

We now consider the terms involving the expressions $\beta^{2}u_{j}$ and $\displaystyle\frac{1}{\sqrt{n}}W_{j}$. By the mean-value theorem,

[TABLE]

By tightness of the field, we then get

[TABLE]

The convergence of the terms involving $W_{j}$ instead of $u_{j}$ follows by comparison as

[TABLE]

Hence, all linear terms appearing in the previous section converge to transport terms.

7.3. Martingale term

The quadratic variation of the martingale part satisfies

[TABLE]

By a criterion of Aldous [4], this implies convergence to the white noise.

7.4. Symmetric term

Recall that

[TABLE]

The argument used to treat the linear terms in Section 7.2 immediately shows that

[TABLE]

7.5. Anti-symmetric term

All that is left is to identify the limit of the term $\widetilde{\mathcal{B}}^{n}_{t}$. Define a modified version of the field by

[TABLE]

By careful $L^{2}$ computations,

[TABLE]

so that, when integrated over time, the field and the modified field (and their squares) are equivalent.

Recall $\iota_{\varepsilon}(x)=\varepsilon^{-1}{\bf 1}_{(x,x+\varepsilon]}$ and observe that

[TABLE]

From here, we obtain the limit

[TABLE]

This does not follow immediately from the convergence of the field as $\iota_{\varepsilon}$ is not an $\mathcal{S}(\mathbb{R})$ function. However, it can be approximated by $\mathcal{S}(\mathbb{R})$ functions from where the convergence follows (see [17], Section 5.3).

By Proposition 1,

[TABLE]

With $l\sim\varepsilon\sqrt{n}$ and taking the limit as $n\to\infty$,

[TABLE]

The crucial estimate (EC2) follows from the triangle inequality. By Theorem 2, we get the existence of the limit

[TABLE]

Estimate (6) further yields $\widetilde{\mathcal{B}}=\mathcal{A}$.

We now check that $\mathcal{X}$ satisfies the estimate (EC1). By (6), it is enough to check that

[TABLE]

for all $n\geq 1$. By a summation-by-parts and the smoothness of $\varphi$, it is enough to check that

[TABLE]

This follows at once from Kipnis-Varadhan inequality and the following computation: with $f\in\mathscr{C}$, integration-by-parts and Young’s inequality yield,

[TABLE]

This proves (7).

Finally, we note that all our estimates can be applied to the reversed process $\{\mathcal{X}^{n}_{T-t}:\,t\in[0,T]\}$. This shows that $\mathcal{X}$ satisfies Condition 3 of Definition 3.

Appendix A Estimates on the terms of order 3

The goal of this section is to estimate the term

[TABLE]

We will show that this expression only contributes a few transport terms. The following computations are inspired by [6].

We start with the observation that, in monomials of order 3 (and higher), we can replace each instance of the $u_{j}$’s by the corresponding $W_{j}$’s by paying the price of a term that converges to zero in the ucp topology. For example, a simple $L^{2}$ computation shows that

[TABLE]

Much in the same way, we see that indexes can be shifted. For instance, by a summation by parts, we get

[TABLE]

Any monomial of degree 3 can be treated similarly. Note that indexes can also be shifted in expressions involving $\beta^{2}u_{j}$.

Based on these considerations,

[TABLE]

where

[TABLE]

Putting everything together, we get

[TABLE]

with

[TABLE]

By an $L^{2}$ computation, the linear terms are readily seen to be tight (and to contribute to transport terms in the limit). The terms involving $L$ can be treated with the method of Lemma 4 and 5 and vanish in the limit. The rest of this section is devoted to show that the monomials of order 3 in $W$ can be neglected. It amounts to showing a second-order Boltzmann-Gibbs principle for these terms:

Proposition 2**.**

Let $l\geq 2$. There exists a constant $C>0$ such that

[TABLE]

Proof.

Let

[TABLE]

and use the decomposition

[TABLE]

The first term on the right-hand-side can be treated with a straightforward adaptation of Proposition 1 and gives the bound $\displaystyle C\left(\frac{Tl}{n}+\frac{T^{2}}{l^{2}\sqrt{n}}\right)$. The second term is the object of the next lemma. The remaining terms can be handled by $L^{2}$ computations producing overall the bound $CT^{2}/l^{2}$. ∎

Lemma 6**.**

There exists a constant $C>0$ such that

[TABLE]

Proof.

Let $f\in\mathscr{C}$. Using integration-by-parts,

[TABLE]

where $\psi_{k}=(l-k)/l$. By Young’s inequality, twice the above is bounded by

[TABLE]

Taking $\alpha=2l/\sqrt{n}$, this is further bounded by

[TABLE]

The result follows from Kipnis-Varadhan inequality. ∎

For $T\geq 1/n$ and $l\sim\sqrt{Tn}$, the estimate in Proposition 2 becomes $CT^{3/2}/\sqrt{n}$. On the other hand, a careful $L^{2}$ computation taking dependencies into account yields

[TABLE]

Combining both estimates:

[TABLE]

for $T\geq 1/n$. For $T<1/n$, an $L^{2}$ computation yields

[TABLE]

This allows us to control the centered monomials of order 3. To remove the centering, note that difference $\overline{W}_{j-1}\overline{W}_{j}\overline{W}_{j+1}-W_{j-1}W_{j}W_{j+1}$ only involves terms of the form

[TABLE]

where we used $\mathbb{E}_{n}[W]=O(1/\sqrt{n})$. As we know that the terms of order two are tight, the extra $1/\sqrt{n}$ above makes them to vanish. Finally, the linear terms are tight when normalized by $\sqrt{n}$ and hence vanish with the current normalization.

The other terms of order 3 in (8) are treated similarly.