[Paper Review] Derivation of coupled KPZ equations from interacting diffusions driven by a single-site potential

Kohei Hayashi|arXiv (Cornell University)|Aug 10, 2022

Nonlinear Dynamics and Pattern Formation4 citations

TL;DR

This paper derives coupled KPZ equations from multi-species interacting diffusion processes driven by a single-site nonlinear potential in the high-temperature limit. Using a Taylor expansion to isolate the harmonic part of the potential, the authors establish convergence of fluctuation fields in a moving frame with common characteristic speed, yielding the coupled KPZ equations robustly without assuming weak asymmetry or specific potential forms.

ABSTRACT

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is derived from various microscopic models, and to establish a robust way to derive the KPZ equation is a fundamental problem both in mathematics and in physics. As a microscopic model, we consider multi-species interacting diffusion processes, whose dynamics is driven by a nonlinear potential which satisfies some regularity conditions. In particular, we study asymptotic behavior of fluctuation fields associated with the processes in the high temperature regime under equilibrium. As a main result, we show that when the characteristic speed of each species is the same, the family of the fluctuation fields seen in moving frame with this speed converges to the coupled KPZ equations. Our approach is based on a Taylor expansion argument which extracts the harmonic potential as a main part. This argument works without assuming a specific form of the potential and thereby the coupled KPZ equations are derived in a robust way.

Motivation & Objective

• To establish a rigorous derivation of coupled KPZ equations from microscopic interacting diffusion processes.
• To extend previous results beyond the weakly asymmetric regime, where the anti-symmetric part of the generator is small compared to the symmetric part.
• To develop a method applicable to a broad class of nonlinear potentials without requiring specific functional forms.
• To show convergence of fluctuation fields in a moving frame with common speed to the coupled KPZ system under high-temperature scaling.
• To generalize prior scalar KPZ derivations—such as for the O’Connell-Yor polymer—to the multi-species case with robustness to potential structure.

Proposed method

• Consider a d-species interacting diffusion process driven by a nonlinear potential $V: \mathbb{R}^d \to \mathbb{R}$ satisfying regularity conditions.
• Rescale the potential as $V_\beta(x) = \beta^{-2} V(\beta x)$ and take the high-temperature limit $\beta \to 0$.
• Apply a Taylor expansion around the equilibrium measure to extract the harmonic (quadratic) part of the potential as the dominant contribution.
• Use a moving frame with a common characteristic speed to eliminate macroscopic transport, enabling convergence to the coupled KPZ system.
• Employ a martingale method and spectral gap estimates to control the fluctuation fields and show convergence in distribution.
• Derive the coupling tensor $\Gamma^{i}_{k\ell}$ in the limiting equation through a systematic expansion of the generator and identification of nonlinear terms.

Experimental results

Research questions

• RQ1 Can the coupled KPZ equations be derived from a microscopic model in a strong asymmetric regime where the anti-symmetric and symmetric parts of the generator are on the same order?
• RQ2 Is it possible to derive the coupled KPZ equations without assuming weak asymmetry or a specific form of the potential?
• RQ3 How does the fluctuation field of a multi-species interacting diffusion process behave in the high-temperature limit under a moving frame with common speed?
• RQ4 What is the role of the harmonic part of the potential in the derivation of the KPZ equation in the strong asymmetric regime?
• RQ5 Can the derivation be extended to general nonlinear potentials using only regularity and symmetry assumptions?

Key findings

• The fluctuation fields of the d-species interacting diffusion process, when observed in a moving frame with common characteristic speed, converge to the solution of the coupled KPZ equations.
• The derivation holds for a broad class of nonlinear potentials without requiring weak asymmetry or specific functional forms.
• The limiting equation features a coupling tensor $\Gamma^{i}_{k\ell}$ that arises from the second-order expansion of the potential and the interaction structure.
• The method relies on a Taylor expansion to isolate the harmonic part of the potential, which dominates in the high-temperature limit.
• The convergence is established via a martingale method and spectral gap estimates, ensuring tightness and identification of the limit.
• This work provides the first rigorous derivation of the coupled KPZ equations in a strong asymmetric regime, generalizing prior results on scalar KPZ from models like the O’Connell-Yor polymer.