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

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 $n^{-1\mathrm{/}2}$, 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 $\delta \le 1$ (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.