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

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.