Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911329401438208 |
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| author | Bernardin, Cédric Djurdjevac, Ana Gonçalves, Patricia Schnee, Leander |
| author_facet | Bernardin, Cédric Djurdjevac, Ana Gonçalves, Patricia Schnee, Leander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin Bernardin, Cédric Djurdjevac, Ana Gonçalves, Patricia Schnee, Leander Probability Mathematical Physics Primary 60K35, Secondary 60H15, 60F05 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. |
| title | Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin |
| topic | Probability Mathematical Physics Primary 60K35, Secondary 60H15, 60F05 |
| url | https://arxiv.org/abs/2512.18497 |