A modified Kardar-Parisi-Zhang model

Giuseppe Da Prato,Luciano Tubaro,Arnaud Debussche

doi:10.1214/ecp.v12-1333

Abstract

A one dimensional stochastic differential equation of the form \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] is considered, where $A=\tfrac12 \partial^2_\xi$. The equation is equipped with periodic boundary conditions. When $\alpha=0$ this equation arises in the Kardar-Parisi-Zhang model. For $\alpha\ne 0$, 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 $\alpha>\frac18$.

Highlights

Let us consider the following Burgers equation on (0, 2π) with periodic boundary conditions and perturbed by noise∂ξ2X + ∂ξ(X2) dt + ∂ξdW (t)X(0, ξ) = x(ξ) ∈ L20(0, 2π), X(t, 0) = X(t, 2π). (1.1)A modified Kardar–Parisi–Zhang model where W is a cylindrical white noise of the form ∞W (t, ξ) = ek(ξ)βk(t), k=1 where ek (ξ )√1 2π eikξ, k ∈ Z0, Z0 = Z\{0} and (βk(t))k∈Z0 is a family of standard Brownian motions mutually independent in a filtered probability space (Ω, F, (Ft)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 seen as the limit equation of a suitable particle system, see [4]

Summary

Introduction

Let us consider the following Burgers equation on (0, 2π) with periodic boundary conditions and perturbed by noise. 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). 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 (−A)−α∂ξ [((−A)−α.

Notation

Definition of :X2

Local existence

Global existence