A MODIFIED KARDAR–PARISI–ZHANG MODEL

A one dimensional stochastic diﬀerential equation of the form is considered, where A = 12 ∂ 2 ξ . The equation is equipped with periodic boundary conditions. When α = 0 this equation arises in the Kardar–Parisi–Zhang model. For α 6 = 0, this equation conserves two important properties of the Kardar–Parisi–Zhang model: it contains a quadratic nonlinear term and has an explicit invariant measure which is gaussian. However, it is not as singular and using renormalization and a ﬁxed point result we prove existence and uniqueness of a strong solution provided α > 18 .


Introduction
Let us consider the following Burgers equation on (0, 2π) with periodic boundary conditions and perturbed by noise X(0, ξ) = x(ξ) ∈ L 2 0 (0, 2π), X(t, 0) = X(t, 2π). (1.1) where W is a cylindrical white noise of the form where e k (ξ) = 1 √ 2π e ikξ , k ∈ Z 0 , Z 0 = Z\{0} and (β k (t)) k∈Z0 is a family of standard Brownian motions mutually independent in a filtered probability space (Ω, F , (F t ) t≥0 , P). Equation (1.1) is known as the Kardar-Parisi-Zhang equation (KPZ equation) and was introduced in [15] as a model of the interface growing in the phase transitions theory. It can also seen as the limit equation of a suitable particle system, see [4]. As usual, we write equation (1.1) in an abstract form. It is no restriction to assume that the initial data has a zero average. Since this property is conserved by equation ( where W A (t) is the stochastic convolution (see [12]) Note that (e k ) k∈Z0 is a basis of eigenvectors of A. W A (t) is a Gaussian random variable in L 2 0 (0, 2π) and covariance operator An important characteristic of this problem is that (as it happens for the 2D periodic Navier-Stokes equation), though the problem is non linear, its invariant measure coincides with the Gaussian invariant measure of the corresponding free system, whose covariance operator reduces to the identity in our case. Consequently, the invariant measure does not live in L 2 0 (0, 2π). It is not difficult to see that this measures lives in functional spaces of negative regularity, strictly less than 1/2. A natural way to define the product in this context is to replace the nonlinear term ∂ ξ (X 2 ) by ∂ ξ (: X 2 :), where : X 2 : represents the Wick product. In the case of periodic boundary conditions, the Wick product is the standard product renormalized by the subtraction of an infinite constant. Thus the two products are in fact formally equal since the infinite constant disappears by differentiation. This method based on renormalization has been successfully used recently for some reaction-diffusion equations arising in field theory, see [2], [5], [9], [10], [11], [14], [16] and for 2D-Navier-Stokes equations, see [1], [8], [13]. The case of the Navier-Stokes is very similar to the case considered here. Indeed, the Wick nonlinearity is formally equal to the original nonlinearity. The KPZ equation is more difficult and it is not possible to define the Wick product in the classical way (see [10] for a discussion). A generalized Wick product has been introduced in [3], however it is very irregular and up to now it has not been possible to construct solutions of the KPZ equation with this generalized Wick product. In this article, we adopt another strategy. As it has been done in the case of the stochastic quantization equation, we modify the equation in such a way that the nonlinear term has the same structure and that the equation has the same invariant measure as the KPZ equation. For this reason we shall introduce the following modified equation trying to choose α > 0 as small as possible. It is not difficult to see that indeed the Gaussian measure with covariance equal to the identity is formally invariant for (1.4). It is convenient to introduce a new variable X α (t) = (−A) −α X(t) and to replace the quadratic term X 2 α with the renormalized power :X 2 α : -which just differs from X 2 α by an infinite constant. So, equation (1.4) becomes With this transformation, the invariant measure has now the covariance given by (−A) −2α . For α > 1/4, this measures lives in L 2 0 (0, 2π), it is not necessary to use the Wick product and this equation can be solved by standard arguments. We shall show that the Wick power is well defined provided So, we shall choose α ∈ ( 1 8 , 1 4 ]. Using the strategy introduced in [8,9], we will construct strong solutions for this equation by a suitable fixed point. Using the fact that we know explicitly the invariant measures, we will also prove that the solutions are almost surely global in time. We think that this work is a step in the understanding of the KPZ model and hope that our techniques will generalize so that we can treat the original case α = 0.

Notation
Let us consider the Hilbert space x(ξ)y(ξ)dξ, x, y ∈ H, and the associated norm denoted by | · |. A complete orthonormal system in H is given by and Z 0 = Z\{0}. It is well known that these are the eigenvectors of A: We set In the following we shall identify H with 2 (Z 0 ) through the isomorphism: We set H := R ∞ so that H is identified as a subspace of H . We denote by µ the product measure on H We also use the L 2 based Sobolev spaces which in our case are easily characterized thanks to the eigenbasis of A. For s ∈ R, we set Equations (1.4) and (2.1) are equivalent. We consider only (2.1) without mentioning the corresponding results for equation (1.4).
In order to lighten the notations, we omit the subscript α and below X denotes the unknown of equation (2.1). Since we work only with this equation, this should not yield any confusion. We denote by µ α the Gaussian measure (corresponding to the free system) Below, we shall use the following well known result in finite dimensions: given the following system of SDE (free system) where Proof. It is clear that for any x ∈ P N H Moreover, for x ∈ P N H, It follows and Then this fact, together with (2.3), implies that µ N = N 0, (−A N ) −2α is invariant for (2.2).

Definition of :X 2 :
Let us recall the definition of Wick product :X 2 : in our specific case following the method of [10]. We denote by e k , · the k th coordinate mapping defined on H and we set for X ∈ H X N (ξ) = 1≤|k|≤N e k , X e k (ξ) and : where (2.5) Clearly, for any X ∈ H , :X 2 N : is an element of H and therefore of H s for any s ≤ 0. The following result is proved in [10], section 7.
Theorem 2.2. If 1 8 < α ≤ 1 4 then the sequence of functions (:X 2 N :) has a limit in L 2 (H −ε , µ α ; H −ε ), for any ε > 1 2 (1 − 4α). We denote this limit by :X 2 :. Unfortunately, the definition of the Wick product is much more complicated for α < 1 8 . It is defined only in a space of generalized random variables (see [3]) and we are not able to handle it. Thus, we shall restrict ourselves from now on to the case α ∈ ( 1 8 , 1 4 ]. Note that for any N ∈ N We deduce that the following result.
It is therefore natural to consider the equation (2.6) We are now able to define the nonlinear term for a random variable whose law is given by µ α and, proceeding as in [16], this is sufficient to construct a weak stationary solution. We wish to go further and define the nonlinear term for a larger class of random variable. The following result proved by paraproduct techniques (see [6], [7]) is useful.
Consider now a random variable X with values in H which can be written as X = Y + Z where Z has the law µ and Y ∈ L 2 (Ω; H β ). We can write Using Theorem 2.2 and Lemma 2.4, the three terms have a limit in L 2 (Ω; H δ ) provided β > 1 2 − 2α and δ < β − 1 2 + 2α. We are therefore able to define the nonlinear term for such random variables and we have the following natural formula : Finally, if we know only that Y ∈ H β almost surely, the above discussion still holds but the limit has to be understood in probability. Again, :X 2 : is defined through the subtraction of an infinite constant and ∂ ξ [:X 2 :] is a natural definition for the nonlinear term.

Local existence
We write equation (2.6) in the mild form It follows from the factorization method (see [12]) that The following Lemma is proved as in [8], [9].

Lemma 2.5. We have
We will see that Y is regular and thanks to (2.7), (2.8) becomes (2.12) We are going to solve equation (2.12) by a fixed point argument in the space where γ > 0, β > 0 and r ≥ 1 will be chosen later and T is sufficiently small. We need the following lemma.
Moreover, since for all β, γ ∈ R we see that e tA y ∈ L r (0, T ; H β ) provided condition (2.13) is fulfilled.
We deduce

Electronic Communications in Probability
Then by Hausdorff-Young inequality The claim follows. The proof of (iii) is easier and left to the reader.
The following lemma states that the conditions of Lemma 2.6 are compatible.
Proof. Using classical arguments (see the proof of Theorem 5.1 in [8] for details), it suffices to obtain a uniform a priori estimate on the solutions of the Galerkin approximations. We follow here the method in [9], [13]. In order to lighten the notations, we perform directly the computations below on the solutions of equation (2.8). A rigorous proof is easily obtained by translating these computations on the Galerkin solutions. We have X(t, x) = e tA x + 1 Finally, it is not difficult to see, using the factorization method (see [12]), that for µ α -almost all x and then the global existence for µ α -almost all x follows. This ends the proof of Theorem 2.9.