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Jointly stationary solutions of periodic Burgers flow

Alexander Dunlap, Yu Gu

TL;DR

This work analyzes the stochastic Burgers equation on $ ext{T}$ with periodic forcing, establishing a differentiable dependence of the coupled invariant solutions $ ext{U}_ heta$ on the mean velocity parameter $ heta$. By applying the Hopf–Cole transform, the authors derive a random Fokker–Planck equation for the derivative in $ heta$, denoting its stationary density by $ ilde{g}_ heta$, and prove the key identity $ ext{U}_{ heta} - ext{U}_{0} = rac{1}{1} rac{d}{d heta} ext{U}_ heta = ilde{g}_ heta$ integrated in $ heta$, connecting Burgers dynamics to a diffusion in a random environment. They develop a one-force-one-solution principle for the FP equation via probabilistic (polymer) representations, prove exponential mixing and convergence to a global FP solution, and interpret $ ilde{g}_ heta$ as the Radon–Nikodym derivative of the environment seen from the particle. In the spacetime white-noise limit, the derivative structure is described by a Busemann-function framework with explicit Brownian-bridge densities, illustrating a deep link between coupled invariant measures, KPZ/Burgers universality, and the ‘environment seen from the particle’ perspective. The results provide a robust framework for the long-time behavior of a passive tracer in a Burgers flow and highlight the rich structure of parameter-dependent invariant measures in stochastic PDEs on compact domains.

Abstract

For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures and study how they change as the velocity parameter varies. We show that the derivative of the invariant measure with respect to the velocity parameter exists, and it can be interpreted as the steady state of a diffusion advected by the Burgers flow

Jointly stationary solutions of periodic Burgers flow

TL;DR

This work analyzes the stochastic Burgers equation on with periodic forcing, establishing a differentiable dependence of the coupled invariant solutions on the mean velocity parameter . By applying the Hopf–Cole transform, the authors derive a random Fokker–Planck equation for the derivative in , denoting its stationary density by , and prove the key identity integrated in , connecting Burgers dynamics to a diffusion in a random environment. They develop a one-force-one-solution principle for the FP equation via probabilistic (polymer) representations, prove exponential mixing and convergence to a global FP solution, and interpret as the Radon–Nikodym derivative of the environment seen from the particle. In the spacetime white-noise limit, the derivative structure is described by a Busemann-function framework with explicit Brownian-bridge densities, illustrating a deep link between coupled invariant measures, KPZ/Burgers universality, and the ‘environment seen from the particle’ perspective. The results provide a robust framework for the long-time behavior of a passive tracer in a Burgers flow and highlight the rich structure of parameter-dependent invariant measures in stochastic PDEs on compact domains.

Abstract

For the one dimensional Burgers equation with a random and periodic forcing, it is well-known that there exists a family of invariant measures, each corresponding to a different average velocity. In this paper, we consider the coupled invariant measures and study how they change as the velocity parameter varies. We show that the derivative of the invariant measure with respect to the velocity parameter exists, and it can be interpreted as the steady state of a diffusion advected by the Burgers flow
Paper Structure (18 sections, 23 theorems, 199 equations)

This paper contains 18 sections, 23 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Context
  4. Organization of the paper
  5. From Burgers to Fokker-Planck
  6. A one-force-one-solution principle for the Fokker-Planck equation
  7. Probabilistic representation
  8. Moment estimates
  9. Exponential mixing of polymer and Burgers
  10. Proof of Theorem \ref{['t.ofos']}
  11. Proof of Theorems \ref{['t.mainth']} and \ref{['t.envi']}
  12. Uniqueness
  13. Environment seen from the particle
  14. The spacetime white noise
  15. Burgers equation with periodic forcing
  16. ...and 3 more sections

Key Result

Theorem 1.1

(i) For any $\theta\in\mathbb{R}$, there exists a unique positive global solution $\tilde{g}_\theta$ to that is spacetime stationary and satisfies $\mathbf{E}\, \tilde{g}_\theta(0,0)=1$, $\tilde{g}_\theta(0,0)\in L^p(\Omega)$ for any $p\in [1,\infty)$, and, with probability $1$, $\int_{\mathbb{T}} \tilde{g}_\theta(t,x)dx=1$ for each $t\in\mathbb{R}$. (ii) For any $\lambda\in\mathbb{R}$ and $t\in\

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 16 more