Extension to boundary mechanisms inducing non-linear Robin conditions

Extend the fluctuation analysis for the stirring process with births and deaths to boundary mechanisms that drive the hydrodynamic limit to the heat equation with non-linear Robin boundary conditions, such as the exclusion processes with non-reversible boundary recently studied by Landim, Mangi, and Salvador. Derive the corresponding non-equilibrium fluctuation limits and identify the associated martingale problem and operator domains.

Background

The current paper establishes fluctuations for a model whose hydrodynamic limit has (linear) Robin boundary conditions determined by the boundary dynamics. Other boundary mechanisms lead to non-linear Robin conditions at the macroscopic level, notably in exclusion processes with non-reversible boundary.

Adapting the fluctuation framework—including test function spaces, martingale characterizations, and correlation bounds—to these non-linear boundary settings is an open extension.

References

We conclude by collecting a few interesting problems, closely related to the ones investigated in this article, that remain open for further study. A third line of investigation concerns the extension of our methods to other boundary mechanisms. For instance, it would be interesting to treat boundary dynamics that drive the system to the heat equation with non-linear Robin boundary conditions, such as in the model recently studied in .

— Non-equilibrium fluctuations for the stirring process with births and deaths  (2512.12902 - Birmpa et al., 15 Dec 2025) in Introduction, Future problems