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

TL;DR

This paper studies how microscopic boundary perturbations influence the macroscopic behavior of stochastic Burgers equations, revealing a critical coupling strength that determines the emergence of boundary conditions in the limit.

Contribution

It introduces a coupled heat bath perturbation and characterizes the resulting boundary conditions in the stochastic Burgers equation at different coupling regimes.

Findings

In the strong-coupling regime, the fluctuation field converges to an SBE with a Dirichlet boundary condition.

In the weak-coupling regime, the fluctuations converge to the standard SBE on the full line without boundary conditions.

Identifies a critical scaling of the coupling strength that governs the boundary behavior in the limit.

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 \(n^{-\delta}\) acting on the particle system. We prove that for \(\delta \leq 1\) (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 \(\delta > 1\) (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 \(\delta\), which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.