TY - JOUR

T1 - A coupled KPZ equation, its two types of approximations and existence of global solutions

AU - Funaki, Tadahisa

AU - Hoshino, Masato

N1 - Funding Information:
The first author is supported in part by the JSPS KAKENHI Grant Numbers (S) 24224004, (S) 16H06338, (B) 26287014 and 26610019. The second author is supported by JSPS KAKENHI, Grant-in-Aid for JSPS Fellows, 16J03010.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - 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.

AB - 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.

KW - KPZ equation

KW - Paracontrolled calculus

KW - Renormalization

KW - Stochastic partial differential equation

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U2 - 10.1016/j.jfa.2017.05.002

DO - 10.1016/j.jfa.2017.05.002

M3 - Article

AN - SCOPUS:85019405610

VL - 273

SP - 1165

EP - 1204

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -