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Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin

Cédric Bernardin, Ana Djurdjevac, Patricia Gonçalves, Leander Schnee

TL;DR

This work rigorously derives the equilibrium fluctuations of a Hamiltonian oscillator chain under a moving heat-bath perturbation and identifies a sharp coupling scaling where a macroscopic Dirichlet boundary condition at the origin emerges in the stochastic Burgers equation. By introducing and exploiting the energy-solution framework for SPDEs, the authors show that in the strong-coupling regime (small delta) the limiting fluctuation field solves SBE on the line with Dirichlet boundary, while in the weak-coupling regime (large delta) the limit reduces to the full-line SBE (Ornstein–Uhlenbeck when the nonlinearity vanishes). The analysis hinges on precise test-function spaces (S, S0, SDir) and a second-order Boltzmann–Gibbs principle to handle nonlinear terms, plus tightness and uniqueness results that tie the microscopic model to macroscopic SPDE descriptions. The results illuminate how boundary conditions can emerge or disappear from microscopic reservoirs in infinite volume, contributing to the broader understanding of KPZ universality and stochastic boundary-driven systems.

Abstract

We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on \(\mathbb{Z}\) with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} \(t \mapsto t n^2\) and an interaction strength tuned by \(n^{-1/2}\), the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength \(n^{-δ}\) acting on the particle system. We prove that for \(δ\leq 1\) (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for \(δ> 1\) (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength \(δ\), which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.

Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin

TL;DR

This work rigorously derives the equilibrium fluctuations of a Hamiltonian oscillator chain under a moving heat-bath perturbation and identifies a sharp coupling scaling where a macroscopic Dirichlet boundary condition at the origin emerges in the stochastic Burgers equation. By introducing and exploiting the energy-solution framework for SPDEs, the authors show that in the strong-coupling regime (small delta) the limiting fluctuation field solves SBE on the line with Dirichlet boundary, while in the weak-coupling regime (large delta) the limit reduces to the full-line SBE (Ornstein–Uhlenbeck when the nonlinearity vanishes). The analysis hinges on precise test-function spaces (S, S0, SDir) and a second-order Boltzmann–Gibbs principle to handle nonlinear terms, plus tightness and uniqueness results that tie the microscopic model to macroscopic SPDE descriptions. The results illuminate how boundary conditions can emerge or disappear from microscopic reservoirs in infinite volume, contributing to the broader understanding of KPZ universality and stochastic boundary-driven systems.

Abstract

We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} and an interaction strength tuned by , the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength acting on the particle system. We prove that for (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength , which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.

Paper Structure

This paper contains 23 sections, 27 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model and main results
  3. Description of the model
  4. Space of test functions
  5. Fluctuation field
  6. Stochastic Partial Differential Equations
  7. Main results
  8. Existence and Characterization of limit-points
  9. Characterization of limit-points
  10. The martingale problem
  11. Convergence of the martingale to a continuous Gaussian process
  12. Additional boundary conditions when $\delta<1$.
  13. Tightness
  14. Uniqueness in law
  15. Uniqueness for SBE($\mathcal{S}_{\text{Dir}}$)
  16. ...and 8 more sections

Key Result

Proposition 2.1

There exists a measurable subset $\Omega \subset (0, \infty)^{\mathbb Z}$ such that for any $\beta>0, \lambda>-1$, $\nu_{\beta, \lambda} (\Omega)=1$, where $\nu_{\beta, \lambda}$ was defined in def:nu. For any initial condition $\zeta_0 \in \Omega$, there exists a unique ${\mathcal{B}}_t$-adapted ti

Figures (1)

  • Figure 1: Illustration of the main theorem regarding the space of test functions and respective equations.

Theorems & Definitions (63)

  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3: Fluctuation Field
  • Remark 2.4
  • Definition 2.5: Energy solutions
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 53 more