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Ergodicity and synchronization of the Kardar-Parisi-Zhang equation

Christopher Janjigian, Firas Rassoul-Agha, Timo Seppäläinen

TL;DR

The paper establishes a comprehensive ergodic and synchronization theory for the 1+1 dimensional KPZ equation via a novel Busemann process that couples all Brownian-invariant stationary solutions. By linking KPZ to the stochastic heat equation and its continuum directed polymer representation, it proves a one force–one solution principle (1F1S) in a non-compact, rough-forcing setting, identifies an (almost surely) empty or dense set of exceptional slopes where synchronization may fail, and constructs pullback attractors that synchronize across initial conditions. It also proves a locally uniform shape theorem leading to stochastic homogenization with an explicit effective Hamiltonian, and establishes ergodicity for both SHE and KPZ stationary measures. The framework unifies stationary distributions, Busemann limits, and semi-infinite polymers, yielding deep insights into the ergodic and homogenized behavior of KPZ and its continuum polymer representation. The results advance the KPZ universality program by providing a robust, fully continuous, non-compact, rough-noise 1F1S theory and connecting microscopic stochastic dynamics to macroscopic homogenized behavior through correctors.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a one force--one solution (1F1S) principle or synchronization: the solution to the KPZ equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift, which shows in particular that these invariant measures are totally ergodic. Our proof constructs the Busemann process for the equation, which gives the natural jointly stationary coupling of all of these stationary solutions. Synchronization then holds simultaneously (on a single full probability event) for all but an at most countable random set of asymptotic slopes. This set of exceptional slopes of instability for which synchronization fails is either almost surely empty or almost surely dense. Along the way, we prove a shape theorem which implies almost sure stochastic homogenization of the KPZ equation, for which the Busemann process gives the process of correctors. We also show that the forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite continuum polymers whose transitions are Doob transforms via Busemann functions of the transitions of the finite length polymers.

Ergodicity and synchronization of the Kardar-Parisi-Zhang equation

TL;DR

The paper establishes a comprehensive ergodic and synchronization theory for the 1+1 dimensional KPZ equation via a novel Busemann process that couples all Brownian-invariant stationary solutions. By linking KPZ to the stochastic heat equation and its continuum directed polymer representation, it proves a one force–one solution principle (1F1S) in a non-compact, rough-forcing setting, identifies an (almost surely) empty or dense set of exceptional slopes where synchronization may fail, and constructs pullback attractors that synchronize across initial conditions. It also proves a locally uniform shape theorem leading to stochastic homogenization with an explicit effective Hamiltonian, and establishes ergodicity for both SHE and KPZ stationary measures. The framework unifies stationary distributions, Busemann limits, and semi-infinite polymers, yielding deep insights into the ergodic and homogenized behavior of KPZ and its continuum polymer representation. The results advance the KPZ universality program by providing a robust, fully continuous, non-compact, rough-noise 1F1S theory and connecting microscopic stochastic dynamics to macroscopic homogenized behavior through correctors.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to admit Brownian motion with a linear drift as a stationary distribution (modulo additive constants). We show that these solutions are attractive, a result known as a one force--one solution (1F1S) principle or synchronization: the solution to the KPZ equation started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift, which shows in particular that these invariant measures are totally ergodic. Our proof constructs the Busemann process for the equation, which gives the natural jointly stationary coupling of all of these stationary solutions. Synchronization then holds simultaneously (on a single full probability event) for all but an at most countable random set of asymptotic slopes. This set of exceptional slopes of instability for which synchronization fails is either almost surely empty or almost surely dense. Along the way, we prove a shape theorem which implies almost sure stochastic homogenization of the KPZ equation, for which the Busemann process gives the process of correctors. We also show that the forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite continuum polymers whose transitions are Doob transforms via Busemann functions of the transitions of the finite length polymers.
Paper Structure (40 sections, 73 theorems, 413 equations)

This paper contains 40 sections, 73 theorems, 413 equations.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Ergodicity, one force--one solution, and pullback attraction
  4. Busemann process
  5. Solving the KPZ equation and our coupling
  6. Integrable inputs and the shape theorem
  7. Stochastic homogenization
  8. Notation and conventions
  9. Solutions of SHE and KPZ
  10. Probability space
  11. Solving SHE
  12. Physical solutions of KPZ
  13. Continuum directed polymer
  14. Main results
  15. Busemann process
  16. ...and 25 more sections

Key Result

Theorem 2.2

There exists a process $Z=\{Z(t,y\space\vert\space s, x): (s, x, t, y)\in\mathop{\mathrm{\mathbb{R}}}\nolimits_{{\scaleobj{0.8}{\uparrow}}}^4\}$ taking values in $\mathop{\mathrm{\mathcal{C}}}\nolimits(\mathop{\mathrm{\mathbb{R}}}\nolimits_{{\scaleobj{0.8}{\uparrow}}}^4,\mathop{\mathrm{\mathbb{R}}}\

Theorems & Definitions (164)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Remark 3.6
  • ...and 154 more