# Regularization by noise and stochastic Burgers equations

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## Abstract

### Keywords

Kardar–Parisi–Zhang equation SPDEs Noise regularization### Mathematics Subject Classification (2000)

00X00 35R60 82B41The main difficulty with Eq. (1) is given by the rough nonlinearity which is incompatible with the distributional nature of the typical trajectories of the process. Note in fact that, at least formally, Eq. (1) preserves the white noise on \(H\) and that the square in the non-linearity is almost surely \(+\infty \) on the white noise. Additive renormalizations in the form of Wick products are not enough to cure this singularity [9].

Jara and Gonçalves [15] introduced a notion of *energy solution* for Eq. (1) and showed that the macroscopic current fluctuations of a large class of weakly non-reversible particle systems on \(\mathbb{Z }\) obey the Burgers equation in this sense. Moreover their results show that also the Hopf–Cole solution is an energy solution of Eq. (1).

More recently Hairer [18] obtained a complete existence and uniqueness result for KPZ. In this remarkable paper the theory of controlled rough paths is used to give meaning to the nonlinearity and a careful analysis of the series expansion of the candidate solutions allow to give a consistent meaning to the equation and to obtain a uniqueness result. In particular Hairer’s solution coincide with the Cole–Hopf ansatz.

In this paper we take a different approach to the problem. We want to point out the regularizing effect of the linear stochastic part of the equation on the the non-linear part. This is linked to some similar remarks of Assing [3, 4] and by the approach of Jara and Gonçalves [15]. Our point of view is motivated also by similar analysis in the PDE and SPDE context where the noise or a dispersive term provide enough regularization to treat some non-linear term: there are examples involving the stochastic transport equation [12], the periodic Korteweg-de Vries equation [5, 17] and the fast rotating Navier–Stokes equation [6]. In particular in the paper [17] it is shown how, in the context of the periodic Korteweg-de Vries equation, an appropriate notion of controlled solution can make sense of the non-linear term in a space of distributions. This point of view has also links with the approach via controlled paths to the theory of rough paths [16].

*small time*behaviour similar to that of the stationary Ornstein–Uhlenbech process \(X\) which solves the linear part of the dynamics:

*time integral*of the non-linear term appearing in SBE\(_\theta \) is well defined, namely that for all \(v\in \mathcal R _\theta \)

The existence of the drift process (6) allows to formulate naturally the SBE\(_\theta \) equation in the space \(\mathcal R _\theta \) of controlled processes and gives a notion of solution quite similar to that of energy solution introduced by Jara and Gonçalves [15]. Existence of (probabilistically) weak solutions will be established for any \(\theta > 1/2\), that is well below the KPZ regime. The precise notion of solution will be described below. We are also able to show easily pathwise uniqueness when \(\theta > 5/4\) but the case \(\theta =1\) seems still (way) out of range for this technique. In particular the question of pathwise uniqueness is tightly linked with that of existence of strong solutions and the key estimates which will allow us to handle the drift (6) are not strong enough to give a control on the difference of two solutions (with the same noise) or on the sequence of Galerkin approximations.

Similar regularization phenomena for stochastic transport equations are studied in [12] and in [10] for infinite dimensional SDEs. This is also linked to the fundamental paper of Kipnis and Varadhan [21] on CLT for additive functionals and to the Lyons–Zheng representation for diffusions with singular drifts [13, 14, 23].

**Plan** In Sect. 1 we define the class of controlled paths and we recall some results of the stochastic calculus via regularization which are needed to handle the Itô formula for the controlled processes. Section 2 is devoted to introduce our main tool which is a moment estimate of an additive functional of a stationary Dirichlet process in terms of the quadratic variation of suitable forward and backward martingales. In Sect. 3 we use this estimate to provide uniform bounds for the drift of any stationary solution. These bounds are used in Sect. 4 to prove tightness of the approximations when \(\theta > 1/2\) and to show existence of controlled solution of the SBE via Galerkin approximations. Finally in Sect. 5 we prove our pathwise uniqueness result in the case \(\theta > 5/4\). In Sect. 6 we discuss related results for the model introduced in [9].

**Notations** We write \(X \lesssim _{a,b,\ldots } Y\) if there exists a positive constant \(C\) depending only on \(a,b,\ldots \) such that \(X \le C Y\). We write \(X \sim _{a,b,\ldots } Y\) iff \(X\lesssim _{a,b,\ldots } Y \lesssim _{a,b,\ldots } X\).

We let \(\mathcal S \) be the space of smooth test functions on \(\mathbb{T }\), \(\mathcal S ^{\prime }\) the space of distributions and \(\langle \cdot ,\cdot \rangle \) the corresponding duality.

On the Hilbert space \(H={L^2_0(\mathbb{T })}\) the family \(\{e_k\}_{k\in \mathbb{Z }_0}\) is a complete orthonormal basis. On \(H\) we consider the space of smooth cylinder functions \(\mathcal C yl\) which depends only on finitely many coordinates on the basis \(\{e_k\}_{k\in \mathbb{Z }_0}\) and for \(\varphi \in \mathcal C yl\) we consider the gradient \(D \varphi : H\rightarrow H\) defined as \(D \varphi (x) = \sum _{k\in \mathbb{Z }_0} D_k \varphi (x) e_k\) where \(D_k = \partial _{x_k}\) and \(x_k = \langle e_k,x \rangle \) are the coordinates of \(x\).

Denote \(\mathcal{C }_T V = C([0,T],V)\) the space of continuous functions from \([0,T]\) to the Banach space \(V\) endowed with the supremum norm and with \(\mathcal{C }^\gamma _T V = C^\gamma ([0,T],V)\) the subspace of \(\gamma \)-Hölder continuous functions in \(\mathcal{C }_T V\) with the \(\gamma \)-Hölder norm.

## 1 Controlled processes

**Definition 1**

*Controlled process*) For any \(\theta \ge 0\) let \(\mathcal R _\theta \) be the space of stationary stochastic processes \((u_t)_{0 \le t \le T}\) with continuous paths in \(\mathcal S ^{\prime }\) such that

- i)
the law of \(u_t\) is the white noise \(\mu \) for all \(t\in [0,T]\);

- ii)there exists a process \(\mathcal{A }\in C([0,T],\mathcal S ^{\prime })\) of zero quadratic variation such that \(\mathcal{A }_0 = 0\) and satisfying the equationfor any test function \(\varphi \in \mathcal S \), where \(M_t(\varphi )\) is a martingale with respect to the filtration generated by \(u\) with quadratic variation \([M(\varphi )]_t = 2t\Vert A^{\theta /2} \varphi \Vert _{L^2_0(\mathbb{T })}^2\);$$\begin{aligned} u_t(\varphi ) = u_0(\varphi ) + \int \limits _0^t u_s(-A^\theta \varphi ) \mathrm{d }s+\mathcal{A }_t(\varphi ) + M_t(\varphi ) \end{aligned}$$(7)
- iii)
the reversed processes \(\hat{u}_t = u_{T-t}\), \(\hat{\mathcal{A }}_t = -\mathcal{A }_{T-t}\) satisfies the same equation with respect to its own filtration (the backward filtration of \(u\)).

For controlled processes we will prove that if \(\theta >1/2\) the Burgers drift is well defined by approximating it and passing to the limit. Let \(\rho :\mathbb{R }\rightarrow \mathbb{R }\) be a positive smooth test function with unit integral and \(\rho ^{\varepsilon }(\xi )=\rho (\xi /{\varepsilon })/{\varepsilon }\) for all \({\varepsilon }>0\). For simplicity in the proofs we require that the function \(\rho \) has a Fourier transform \(\hat{\rho }\) supported in some ball and such that \(\hat{\rho }= 1\) in a smaller ball. This is a technical condition which is easy to remove but we refrain to do so here not to obscure the main line of the arguments.

**Lemma 1.**

*denote*with \(\int _0^t F( u_s) \mathrm{d }s\) the resulting process with values in \(C([0,T],\mathcal{F }L^{\zeta ,\infty })\).

*Proof*

We postpone the proof in Sect. 3. \(\square \)

It will turn out that for this process we have a good control of its space and time regularity and also some exponential moment estimates. Then it is relatively natural to *define* solutions of Eq. (4) by the following self-consistency condition.

**Definition 2**

*Controlled solution*) Let \(\theta >1/2\), then a process \(u\in \mathcal R _\theta \) is a controlled solution of SBE\(_\theta \) if almost surely

Note that these controlled solutions are a generalization of the notion of probabilistically weak solutions of SBE\(_\theta \). The key point is that the drift term is not given explicitly as a function of the solution itself but characterized by the self-consistency relation (8). In this sense controlled solutions are to be understood as a couple \((u,\mathcal A )\) of processes satisfying compatibility relations.

An analogy which could be familiar to the reader is that with a diffusion on a bounded domain with reflected boundary where the solution is described by a couple of processes \((X,L)\) representing the position of the diffusing particle and its local time at the boundary [22].

Note also that there is no requirement on \(\mathcal A \) to be adapted to \(u\). Our analysis below cannot exclude the possibility that \(\mathcal A \) contains some further randomness and that the solutions are strictly weak, that is not adapted to the filtration generated by the martingale term and the initial condition.

## 2 The Itô trick

**Lemma 2**

*Proof*

The bound (11) in the present form (with the use of the backward martingale to remove the drift part) has been inspired by [8, Lemma 4.4].

**Lemma 3**

*Proof*

## 3 Estimates on the Burgers drift

**Lemma 4**

*Proof*

Using Lemma 2 and the estimates contained in Lemma 4 we are led to the next set of more refined estimates for the drift and his small scale contributions.

**Lemma 5**

*Proof*

Analogous estimates go through also for the functions obtained via convolution with the \(e^{-A^\theta t}\) semi-group.

**Lemma 6**

**Corollary 1**

*Proof*

**Remark 1**

At this point we are in position to prove Lemma 1 on the existence of the Burgers’ drift for controlled processes.

*Proof*

## 4 Existence of controlled solutions

**Lemma 7**

*Proof*

We are now ready to prove our main theorem on existence of (probabilistically weak) controlled solutions to the generalized SBE.

**Theorem 1**

*Proof*

## 5 Uniqueness for \(\theta >5/4\)

In this section we prove a simple pathwise uniqueness result for controlled solutions which is valid when \(\theta > 5/4\). Note that to each controlled solution \(u\) is naturally associated a cylindrical Brownian motion \(W\) on \(H\) given by the martingale part of the controlled decomposition (7). Pathwise uniqueness is then understood in the following sense.

**Definition 3**

SBE\(_\theta \) has pathwise uniqueness if given two controlled processes \(u,\tilde{u}\in \mathcal R _\theta \) on the same probability space which generate the same Brownian motion \(W\) and such that \(\tilde{u}_0 = u_0\) amost surely then there exists a negligible set \(\mathcal N \) such that for all \(\varphi \in \mathcal S \) and \(t\ge 0\)\(\{u_t(\varphi ) \ne \tilde{u}_t(\varphi )\} \subseteq \mathcal N \).

**Theorem 2**

The generalized SBE has pathwise uniqueness when \(\theta >5/4\).

*Proof*

## 6 Alternative equations

The technique of the present paper extends straighforwardly to some other modifications of the SBE.

### 6.1 Regularization of the convective term

### 6.2 The Sasamoto–Spohn discrete model

## 7 2D stochastic Navier–Stokes equation

We consider the problem of stationary solutions to the 2d stochastic Navier–Stokes equation considered in [1] (see also [2]). We would like to deal with invariant measures obtained by formally taking the kinetic energy of the fluid and considering the associated Gibbs measure. However this measure is quite singular and we need a bit of hyperviscosity in the equation to make our estimates work.

### 7.1 The setting

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