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