[Paper Review] Ergodicity and synchronization of the Kardar-Parisi-Zhang equation

Christopher Janjigian, Firas Rassoul‐Agha+1 more|arXiv (Cornell University)|Nov 13, 2022

Advanced Mathematical Modeling in Engineering9 citations

TL;DR

This paper establishes the one force–one solution (1F1S) principle for the Kardar-Parisi-Zhang (KPZ) equation on the real line by constructing a Busemann process that couples all stationary solutions. It proves that solutions starting from any initial condition with a given slope converge almost surely to a Brownian motion with that drift, demonstrating synchronization and total ergodicity of invariant measures, with exceptional slopes of instability being either almost surely empty or dense.

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.

Motivation & Objective

• To establish the one force–one solution (1F1S) principle for the KPZ equation on the real line.
• To prove that solutions with a given asymptotic slope converge almost surely to a Brownian motion with that drift, demonstrating synchronization.
• To construct a Busemann process that provides a jointly stationary coupling of all stationary solutions to the KPZ equation.
• To analyze the structure of exceptional slopes where synchronization fails, showing they are either almost surely empty or dense.
• To establish a shape theorem implying almost sure stochastic homogenization of the KPZ equation, with the Busemann process providing correctors.

Proposed method

• Constructs the Busemann process as a jointly stationary coupling of all stationary solutions to the KPZ equation.
• Uses the Busemann process to define correctors for stochastic homogenization of the KPZ equation.
• Applies a shape theorem for shift-covariant cocycles to establish almost sure convergence of polymers to semi-infinite counterparts.
• Employs a random dynamical systems framework to analyze long-time behavior and invariance under time shifts.
• Applies Doob transforms to show convergence of finite-length point-to-line and point-to-point polymers to semi-infinite polymers.
• Uses ergodicity criteria and Markovian coupling arguments to prove uniqueness and almost sure convergence of solutions to stationary measures.

Experimental results

Research questions

• RQ1 Does the KPZ equation exhibit synchronization, such that solutions with a given asymptotic slope converge to a Brownian motion with that drift?
• RQ2 What is the structure of the set of exceptional slopes for which synchronization fails?
• RQ3 Can a Busemann process be constructed that couples all stationary solutions in a jointly stationary manner?
• RQ4 Does the KPZ equation admit almost sure stochastic homogenization, and what role do the Busemann functions play in the corrector process?
• RQ5 How do finite-length continuum polymers converge to semi-infinite polymers under the Doob transform via Busemann functions?

Key findings

• Solutions to the KPZ equation with a given asymptotic slope converge almost surely to a Brownian motion with that drift, establishing the 1F1S principle.
• The set of exceptional slopes for which synchronization fails is either almost surely empty or almost surely dense.
• The Busemann process provides a jointly stationary coupling of all stationary solutions, enabling simultaneous synchronization for all but countably many slopes.
• A shape theorem holds for the KPZ equation, implying almost sure stochastic homogenization with the Busemann process serving as the corrector.
• Forward and backward point-to-point and point-to-line continuum polymers converge to semi-infinite polymers whose transitions are Doob transforms via Busemann functions.
• The invariant measures of the KPZ equation are totally ergodic, as shown through the uniqueness and Markovian structure of the Busemann process.