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Stationary solutions to the stochastic Burgers equation on the line

Alexander Dunlap, Cole Graham, Lenya Ryzhik

TL;DR

This work analyzes the stochastic Burgers equation on $\mathbb{R}$ forced by the gradient of a smooth-in-space, white-in-time Gaussian field, focusing on space-time invariant measures. By decomposing the solution into a linear stochastic convolution $\psi$ and a deterministic remainder $\theta$, the authors develop a PDE-based framework in sublinear weighted spaces, establish well-posedness, Feller properties, and periodic approximations, and link to KPZ via the Cole–Hopf transform. They prove the existence, uniqueness, and full classification of extremal (indecomposable) space-time stationary measures $\nu_{\mathbf{a}}$ parameterized by the mean $\mathbf{a}$, and show convergence of evolution from perturbations to these extremal laws. The paper also provides a stable attractor theory: time-evolved processes from bounded or periodic initial data converge in law to the corresponding extremal invariant measure, with ordering and invariance properties underpinning the classification. Together, these results illuminate the long-time behavior, ergodic structure, and KPZ-related aspects of stochastic Burgers on the line and contribute to understanding KPZ universality in this setting.

Abstract

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity.

Stationary solutions to the stochastic Burgers equation on the line

TL;DR

This work analyzes the stochastic Burgers equation on forced by the gradient of a smooth-in-space, white-in-time Gaussian field, focusing on space-time invariant measures. By decomposing the solution into a linear stochastic convolution and a deterministic remainder , the authors develop a PDE-based framework in sublinear weighted spaces, establish well-posedness, Feller properties, and periodic approximations, and link to KPZ via the Cole–Hopf transform. They prove the existence, uniqueness, and full classification of extremal (indecomposable) space-time stationary measures parameterized by the mean , and show convergence of evolution from perturbations to these extremal laws. The paper also provides a stable attractor theory: time-evolved processes from bounded or periodic initial data converge in law to the corresponding extremal invariant measure, with ordering and invariance properties underpinning the classification. Together, these results illuminate the long-time behavior, ergodic structure, and KPZ-related aspects of stochastic Burgers on the line and contribute to understanding KPZ universality in this setting.

Abstract

We consider invariant measures for the stochastic Burgers equation on , forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each , there is a unique indecomposable law of a spacetime-stationary solution with mean , in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean- periodic functions converge in law to the extremal space-time stationary solution with mean as time goes to infinity.

Paper Structure

This paper contains 29 sections, 56 theorems, 431 equations.

Table of Contents

  1. Introduction
  2. Solutions to the Burgers equation in weighted spaces
  3. From an SPDE to a PDE
  4. Solutions in weighted function spaces
  5. The periodized problem
  6. Mild solutions
  7. Local-in-time existence for the periodized problem
  8. Global-in-time existence for the periodized problem
  9. Solutions on the whole space
  10. Comparison principle and L¹-contraction
  11. Pathwise results
  12. Conservation and L¹-contraction in the probability space
  13. Ordering of stationary solutions
  14. Other properties of solutions
  15. Shear-invariance
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $m\in(0,1)$. With probability $1$, there is a map so that for each $v\in\mathcal{X}_m$, $u=\Psi(v)$ is the unique strong solution in $\mathcal{C}_{\mathrm{loc}}([0,\infty);\mathcal{X}_{m})$ to eq:uPDE-1 with $u(0,x)=v(x)$. The map $\Psi$ is continuous almost surely. Finally, the semigroup $P_{t}f(v)=\mathbb{E}[f(u(t,\cdot))]$ has the Feller propert

Theorems & Definitions (110)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3: Feller property
  • proof
  • Lemma 2.4
  • ...and 100 more