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Time-inhomogeneous KPZ equation from non-equilibrium Ginzburg-Landau SDEs

Kevin Yang

TL;DR

The article develops a rigorous framework to derive time-inhomogeneous KPZ-type fluctuations from time-inhomogeneous Ginzburg–Landau SDEs via a Cole–Hopf transform. Central to the approach are a local second-order Boltzmann–Gibbs principle and mesoscopic Yau-type entropy methods, enabling convergence to a time-inhomogeneous KPZ (TIKPZ) with coefficients depending on time. The authors establish global well-posedness for the time-inhomogeneous stochastic heat equation (TISHE) and provide a corollary with an exponential-scale improvement over previous time-homogeneous results. The work introduces homogenization measures, hybrid forward–backward KV inequalities, and a multi-scale probabilistic machinery to handle non-equilibrium fluctuations, offering a blueprint for KPZ-type limits in broader non-stationary particle systems. This framework advances the theoretical understanding of non-equilibrium KPZ universality and offers tools for analyzing singular SPDEs with time-varying coefficients in interacting-particle settings.

Abstract

We introduce a framework, which is a mesoscopic-fluctuation-scale analog of Yau's method [46] for hydrodynamic limits, for deriving KPZ equations with time-dependent coefficients from time-inhomogeneous interacting particle systems. To our knowledge, this is the first derivation of a time-inhomogeneous KPZ equation whose solution theory has an additional nonlinearity that is absent in the time-homogeneous case. So, we also show global well-posedness for the SPDE. To be concrete, we restrict to time-inhomogeneous Ginzburg-Landau SDEs. The method for deriving KPZ is based on a Cole-Hopf transform, whose analysis is the bulk of this paper. The key ingredient for said analysis is a ``local" second-order Boltzmann-Gibbs principle, shown by stochastic calculus of the Ginzburg-Landau SDEs and regularity estimates for their Kolmogorov equations, all of which likely generalizes to many other particle systems. This addresses a ``Big Picture Question" in [47] on deriving KPZ equations. It is also, to our knowledge, a first result on KPZ-type limits in a non-equilibrium like that in [6].

Time-inhomogeneous KPZ equation from non-equilibrium Ginzburg-Landau SDEs

TL;DR

The article develops a rigorous framework to derive time-inhomogeneous KPZ-type fluctuations from time-inhomogeneous Ginzburg–Landau SDEs via a Cole–Hopf transform. Central to the approach are a local second-order Boltzmann–Gibbs principle and mesoscopic Yau-type entropy methods, enabling convergence to a time-inhomogeneous KPZ (TIKPZ) with coefficients depending on time. The authors establish global well-posedness for the time-inhomogeneous stochastic heat equation (TISHE) and provide a corollary with an exponential-scale improvement over previous time-homogeneous results. The work introduces homogenization measures, hybrid forward–backward KV inequalities, and a multi-scale probabilistic machinery to handle non-equilibrium fluctuations, offering a blueprint for KPZ-type limits in broader non-stationary particle systems. This framework advances the theoretical understanding of non-equilibrium KPZ universality and offers tools for analyzing singular SPDEs with time-varying coefficients in interacting-particle settings.

Abstract

We introduce a framework, which is a mesoscopic-fluctuation-scale analog of Yau's method [46] for hydrodynamic limits, for deriving KPZ equations with time-dependent coefficients from time-inhomogeneous interacting particle systems. To our knowledge, this is the first derivation of a time-inhomogeneous KPZ equation whose solution theory has an additional nonlinearity that is absent in the time-homogeneous case. So, we also show global well-posedness for the SPDE. To be concrete, we restrict to time-inhomogeneous Ginzburg-Landau SDEs. The method for deriving KPZ is based on a Cole-Hopf transform, whose analysis is the bulk of this paper. The key ingredient for said analysis is a ``local" second-order Boltzmann-Gibbs principle, shown by stochastic calculus of the Ginzburg-Landau SDEs and regularity estimates for their Kolmogorov equations, all of which likely generalizes to many other particle systems. This addresses a ``Big Picture Question" in [47] on deriving KPZ equations. It is also, to our knowledge, a first result on KPZ-type limits in a non-equilibrium like that in [6].
Paper Structure (125 sections, 64 theorems, 483 equations)

This paper contains 125 sections, 64 theorems, 483 equations.

Table of Contents

  1. Introduction
  2. The aforementioned assumptions for convergence to \ref{['eq:kpz']}
  3. Solving \ref{['eq:kpz']}
  4. About time-inhomogeneity
  5. Some of the main innovations
  6. Cole-Hopf
  7. Second-order Boltzmann-Gibbs principle
  8. Going beyond \ref{['eq:hf']}-\ref{['eq:glsde']}
  9. Acknowledgements
  10. Precise statements for the main results
  11. Time-inhomogeneous KPZ equation
  12. Deriving \ref{['eq:kpz']} from \ref{['eq:hf']}
  13. Homogenization measures
  14. Yau-type assumptions
  15. Main results
  16. ...and 110 more sections

Key Result

Theorem 2.1

Let us first recall eq:she2a-eq:she2b and give the setting. Then, with probability $1$, there exists a unique random process $\mathbf{Z}^{\infty}$ which takes values in $\mathscr{C}([0,\infty)\times\mathbb{T})$ and satisfies eq:she2a-eq:she2b with initial data $\mathbf{Z}^{\infty}(0,\cdot)=\mathbf{Z}^{\infty,\mathrm{in}}(\cdot)$. Moreover, we know $\mathbf{Z

Theorems & Definitions (149)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark
  • Definition 2.4
  • Remark
  • Remark
  • Definition 2.5
  • ...and 139 more