Non-equilibrium fluctuations from non-product initial states (including NESS)

Establish the non-equilibrium fluctuation limit for the one-dimensional stirring process with births and deaths when the system is initialized from non-product measures, in particular from the non-equilibrium stationary state (NESS). Specifically, prove convergence of the density fluctuation field to the appropriate generalized Ornstein–Uhlenbeck process under such non-product initial conditions and develop the necessary correlation (v-function) estimates without relying on product structure.

Background

The paper proves non-equilibrium fluctuations for the stirring process with births and deaths under the crucial assumption that the initial measure is product, which enables sharp estimates on spatial and space–time v-functions used in the martingale characterization.

The authors note that the non-equilibrium stationary state (NESS) for this model is not of product form, so their current results do not cover fluctuations from NESS or other non-product initial conditions. Extending the fluctuation theory beyond product measures requires new techniques to control correlations without propagation of chaos.

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 first natural question concerns the derivation of the non-equilibrium fluctuations for more general initial conditions as, for example, the non-equilibrium stationary state (NESS). It is easy to check that the NESS is not of product form, and given the assumptions that we impose on our initial measure our result does not include the fluctuations from the NESS.

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

Thus, the remaining step toward a path-space central limit theorem is the corresponding tightness and trajectory-regularity estimate.

— Sharp mean-field estimates for diffusive log/Riesz gases in the Hilbert--Schmidt regime  (2609.02743 - Delgadino et al., 2 Sep 2026) in Remark 3.12, “Toward a path-space central limit theorem”; Section 5, subsection “Consequences and open problems”

In the Neumann case, there is an additional contribution coming from the mean of the test function, under the form of the vector $Z$, whose law is not fully characterized. Upon proving that the stationary measure satisfies a central limit theorem for the total number of particles and total energy, one could fully characterize this contribution as a Gaussian random variable and deduce a full convergence result rather than simply a characterization of limit points. One way to prove this would be to estimate the two-point correlation function of the stationary measure, and for this, we cannot apply the strategy of since the equations for the two-point correlation function are not closed. This is a technical difficulty that we leave for future work.

— Dynamical and stationary non-equilibrium fluctuations for a boundary-driven exclusion process with energy  (2609.08701 - Cunha, 8 Sep 2026) in Remark immediately following Theorem “Stationary fluctuations, \(\theta>1\)”, subsection “Hydrostatics and non-equilibrium stationary fluctuations”