A moderate deviations principle for the law of a stochastic Burgers equation is proved via the weak convergence approach. In addition, some useful estimates toward a central limit theorem are established.

We consider the following stochastic Burgers equation with multiplicative space-time white noise, indexed by

The deterministic Burgers equation was introduced in [

Our first goal in this paper is to study the moderate deviations of

The deviation scale

Furthermore, there are basically two approaches to analyzing moderate and large deviations for processes. The former, which is originally used by Freidlin and Wentzell [

We stress here that, in the present paper, we mainly use the weak convergence approach to establish

We also note that the greatest difficulty in studying any aspect of Burgers’ type equations lies in their quadratic term. In fact, most of the techniques usually used to deal with stochastic differential equations with Lipschitz drift coefficients don’t longer work generally, and one resort to localization or tightness argument to circumvent this difficulty.

As pointed out before, we will prove a moderate deviations principle for the stochastic Burgers equation (

The paper is organized as follows. Section

In this paper all positive constants are denoted by

Let

For each

A rigorous meaning to the solution of (

By Theorem 2.1 in [

The deterministic equation (

We now recall some estimations of the Green kernel function

According to Varadhan [

A family of random variables

In the context of the weak convergence approach, the proof of the Laplace principle for functionals of the Brownian sheet is essentially based on the following variational representation formula, which was originally proved in [

Here, we briefly describe the result needed, in our context, for proving the Laplace principle, and state our main result.

Let us first introduce some notations. For

For any positive integer

For

We are now in position to introduce the following result, due to Budhiraja et al. [

In this subsection, we adapt the general scheme described above to study moderate deviations for the equation (

We denote by

This implies (see Theorem IV.9.1. of [

With these notations in mind, the main result of this section is stated in the following

Note that the conclusion of Theorem

We basically follow the same idea as in [

The proof of

For

For the uniqueness, if

The proof follows from a standard fixed point argument, and for the convenience of the reader, we include it in the Appendix. □

Let

We have the following proposition.

Recall that
^{th} summand of the RHS of the above equation.

In view of (

We first consider the cases where

Let

Using the Burkholder–Davis–Gundy inequality, the boundedness of

To deal with

For the tightness of

The proof of the tightness of

To show the tightness of

According to Theorem 2.1 in [

This fact combined with the condition (

Using again the condition (

Thus, the equation (

Now, observe that

Having shown the tightness of each

For

To handle the convergence of each of the other terms, we invoke the Skorohod representation theorem and assume the almost sure convergence on a larger common probability space.

For

For

Using the estimation (

Concerning

The proof of the convergence of

Consequently,

Thus, by the convergence of both the process

Now, let us prove the condition

Let

On one hand, using Lemma

On the other hand, in order to handle the second term in the right hand side of (

Finally, the proof of Theorem

Many results on central limit theorem has been recently established for various kinds of parabolic SPDEs under strong assumptions on the drift coefficient. More specifically, under the linear growth condition, the differentiability and the global Liptschitz condition on both the drift coefficient and its derivative, some central limit theorems have been established in [

We begin with the following result.

We will use similar arguments as in Cardon-Weber and Millet [

Since

Moreover, using the SPDE (

Hence, to prove (

For this purpose, note first that

Now, we can announce and state the following proposition.

We will use a localization argument. For

For

Now, taking the supremum up to time

Notice that

This, together with (

Combining (

Using Gronwall’s lemma we deduce that, for all

Therefore, for any fixed

Letting

Finally, since

This section contains some technical results needed in the proof of the main theorem of the paper.

First, we recall the following result proved in Lemma 3.1 in [

For

The following lemma is a consequence of Lemma 3.1 in [

To use a fixed point argument, we consider, for any given

Step 1. Let

Step 2. Let

The continuity of the solution

In order to prove Lemma

We summarize some important proprieties of the sequence

First notice that since

Therefore by the weak convergence of

Now, let us show (

It remains to prove (

The authors are very thankful to the Editor for her very constructive criticism from which our final version of the article has benefited. Many thanks also to the referees for their careful reading and useful remarks. We are also very indebted to Professors R. Zhang and J. Xiong for some kind discussions we had about the Burkholder–Davis–Gundy inequality for SPDEs driven by a space-time white noise.