A coupled KPZ equation, its two types of approximations and existence of global solutions

TL;DR

This paper studies a multi-component coupled KPZ equation, demonstrating the convergence of two approximation methods, establishing global solutions under certain conditions, and analyzing invariant measures using advanced stochastic calculus techniques.

Contribution

It introduces and compares two approximation schemes for the coupled KPZ equation, proving their convergence and establishing global solutions under the trilinear condition.

Findings

Two approximation methods converge to the same limit under proper renormalization.

The coupled KPZ equation admits a global solution when initial data is from the stationary measure.

The invariant measure is characterized by a twisted Wiener measure.

Abstract

This paper concerns the multi-component coupled Kardar-Parisi-Zhang (KPZ) equation and its two types of approximations. One approximation is obtained as a simple replacement of the noise term by a smeared noise with a proper renormalization, while the other one introduced in [6] is suitable for studying the invariant measures. By applying the paracontrolled calculus introduced by Gubinelli et al. [8, 9], we show that two approximations have the common limit under the properly adjusted choice of renormalization factors for each of these approximations. In particular, if the coupling constants of the nonlinear term of the coupled KPZ equation satisfy the so-called "trilinear" condition, the renormalization factors can be taken the same in two approximations and the difference of the limits of two approximations are explicitly computed. Moreover, under the trilinear condition, the Wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure. This is shown for the associated tilt process. Combined with the strong Feller property shown by Hairer and Mattingly [12], this result can be extended for all initial values.