Non-equilibrium fluctuations for the stirring process with births and deaths

Published 15 Dec 2025 in math.PR | (2512.12902v1)

Abstract: We consider the one-dimensional stirring process on the segment ${-N,\ldots,N}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of size $K$. We investigate the non-equilibrium fluctuations of the system, starting from a product measure associated with a smooth initial profile. Given our initial state, the fluctuations are given by an Ornstein-Uhlenbeck process whose characteristic operators are the Laplacian and gradient operators. The domains of these operators include functions with boundary conditions that depend on the hydrodynamic profile. A central ingredient in our analysis is the derivation of sharp bounds on the space and space-time $v$-functions of arbitrary degree for the centered occupation variables. In particular, we prove that the $v$-functions of degree $2$ and $3$ are of order $N^{-1}$, while those of degree at least $4$ are of order $N^{-1-ζ}$ for some $ζ> 0$.

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Summary

The paper rigorously characterizes fluctuation structures by proving convergence of the rescaled density field to a generalized Ornstein-Uhlenbeck process.
It employs sharp estimates on space and space-time correlation functions to handle non-closed terms from boundary-induced birth and death mechanisms.
The analysis extends hydrodynamic limits from conservative models to complex non-conservative dynamics with memory effects at the boundaries.

Non-Equilibrium Fluctuations in the Stirring Process with Births and Deaths

Introduction and Model Formulation

The paper "Non-equilibrium fluctuations for the stirring process with births and deaths" (2512.12902) rigorously analyzes the fluctuation structure in a boundary-driven stirring (symmetric exclusion) process with non-conservative birth and death dynamics. The system under consideration is one-dimensional, with the bulk implementing a symmetric nearest-neighbor exclusion process and boundaries exhibiting Glauber-type birth and death events on mesoscopic windows of size $K \geq 2$ .

The distinctive feature of this model is the competition between the stirring (which conserves particle number in the bulk) and the boundary injection/removal (which breaks conservation law), tuned to act on macroscopic time scales via accelerated clocks. The mathematical complexity emerges from the interaction between the fast stirring and the conditional, configuration-dependent creation/annihilation rules at the boundary, causing correlations of higher algebraic degree and non-closed evolution equations for local observables.

Hydrodynamic Limit and Macroscopic Description

The authors first recall the hydrodynamic (law of large numbers) behavior: the empirical particle density converges, as $N \to \infty$ , to the solution of the heat equation on $[-1,1]$ with nonlinear Robin-type boundary conditions derived from the microscopic birth/death rules. This result extends standard techniques for conservative stochastic particle systems to non-conservative dynamics with memory effects at the boundary, handled via an auxiliary "factorized" profile that closes the otherwise non-closed evolution equations.

The hydrodynamic limit employs scaling where stirring occurs at rate $N^2$ , and the birth/death process at rate $N$ , making the competition between bulk and boundary effects non-trivial. The resulting limit fulfills:

$\partial_t \rho = \frac 12 \partial^2_u \rho;\quad \partial_u \rho|_{u = \pm 1} = F_\pm(\rho(\pm 1, t)),$

with explicit $F_\pm$ inherited from microscopic rates.

Non-Equilibrium Fluctuations and Martingale Problem

The central question addressed involves understanding stochastic fluctuations around the hydrodynamic evolution: the authors prove that for smooth product-form initial measures, the fluctuation field---the space-time rescaled empirical density, centered and normalized---converges in distribution to a generalized Ornstein-Uhlenbeck (O-U) process defined by the linearized hydrodynamics and quadratic variation inherited from the microscopic model.

The main technical content is the sharp characterization of space and space-time correlation (so-called $v$ -) functions at the mesoscopic and macroscopic scales. These $v$ -functions control the deviation between the stochastic process and its hydrodynamic limit and appear in the higher moments and fluctuation martingale decompositions. Their evolution involves new non-closed terms due to the boundary mechanisms: the configuration-dependent boundary rates induce an increase in the degree of polynomials in occupation variables, nontrivializing closure and tightness arguments.

The fluctuation field $Y_t^\epsilon$ acting on admissible test functions $H$ is defined as:

$Y^\epsilon_t(H) = \sqrt{\epsilon} \sum_{x \in \Lambda_N} H_t(\epsilon x)[\eta_t(x) - \rho_t^\epsilon(x)].$

with $\epsilon = 1/N$ and $\rho_t^\epsilon$ the expectation under the microscopic process. For product-form initial data, the authors establish that the sequence $\{Y^\epsilon_t\}_\epsilon$ converges (in the distributional sense) to the solution of the SPDE:

$\partial_t Y_t = \tfrac12 \partial_u^2 Y_t + \nabla[\sqrt{\chi(\rho_t)} W_t],$

where $\chi(\rho) = \rho(1-\rho)$ and $W_t$ is space-time white noise.

A crucial component for this identification is a martingale problem formulation: the density fluctuation field approximates a sequence of (Dynkin) martingales associated to the O-U process, provided the error terms (expressed in terms of $v$ -functions) are shown to be asymptotically negligible.

Sharp Estimates on Space and Space-Time Correlation Functions

One of the key technical advances is the meticulous bounding of the "truncated" correlation functions, both in space and in space-time. Let $v_n^\epsilon(\underline{x}, t)$ denote the $n$ -th order correlation of centered occupation variables, appropriately renormalized relative to the closed auxiliary profile.

The paper proves strong results:

For $n = 1,2$ , $v_n^\epsilon(\underline{x}, t) = O(\epsilon^{1-2\zeta})$ uniformly over $t$ , $|\underline{x}|$ .
For $3 \le n \le K$ , $v_n^\epsilon = O(\epsilon)$ .
For $n \geq K+1$ , $v_n^\epsilon = O(\epsilon^{1+\zeta})$ for some $\zeta > 0$ .

These bounds rely on integrating the nonlinear Duhamel expansions (containing branching terms from the boundaries), careful analysis using Liggett's inequality and Gaussian kernel estimates for occupation time probabilities, and innovative use of time-integration and stopping-time arguments to gain convergence rates required for tightness.

Analogous (but more involved) estimates are established for the mixed space-time $v$ -functions, which control the time-regularity necessary for proving tightness of the fluctuation field and convergence of the associated martingale problems.

Main Theorem: Fluctuations as Ornstein-Uhlenbeck Process

The main result is the rigorous derivation of the macroscopic fluctuation theory for the stirring process with boundary births and deaths.

Theorem (Non-Equilibrium Fluctuations): Consider the stirring process with the described boundary dynamics, started from a smooth product-form initial measure. The sequence of density fluctuation fields converges in law to a generalized O-U process whose covariance structure is:

$\mathbb{E}[Y_t(H)Y_s(G)] = \sigma(T_t H, T_s G) + \int_0^{s} \langle T_{t-r} H, T_{s-r}G \rangle_{\mathbb L^2(\rho_r)} dr,$

where $T_t$ is the semigroup solving the linearized hydrodynamics with linearized boundary conditions, $\sigma$ is the initial covariance, and the inner product encodes both bulk and boundary quadratic variations.

The proof leverages:

Martingale approximations via Dynkin's formula, including boundary corrections.
Control of error terms via the aforementioned sharp correlation estimates.
Tightness criteria in the Skorokhod space of distributions, using Mitoma's theorem and the Kolmogorov-Centsov criterion.
Identification of the limiting quadratic variation as deterministic, matching the anticipated O-U process.

Implications, Extensions, and Open Problems

From a theoretical perspective, this result substantially fortifies the dynamical theory for non-equilibrium fluctuations in non-conservative interacting particle systems, particularly in boundary-driven models where nontrivial correlations emerge due to the breaking of conservation laws and the complexity of the configuration-dependent rates.

Practically, these techniques and results provide quantitative predictions for fluctuations in non-equilibrium steady states and transient regimes of physical systems modeled by exclusion processes with reservoirs, relevant across statistical mechanics, traffic flow, and interface dynamics.

The authors identify several natural directions for future progress:

Extension of the fluctuation analysis to initial conditions beyond product measures, particularly for the non-equilibrium stationary state (NESS), which is non-product and exhibits even richer correlations.
Systematic study of the fluctuation structure as one varies the relative time scales of bulk and boundary dynamics, including the possibility of slow/fast boundaries and their impact on $v$ -function scaling.
Extension to more complex or nonlinear boundary mechanisms, for instance those inducing genuinely nonlinear Robin conditions as in recent models.

Such extensions will likely require further advances in the control of higher-order correlation functions and possibly new argumentation for closure and tightness, possibly involving entropy or coupling techniques.

Conclusion

This paper provides a comprehensive and highly technical analytic framework for non-equilibrium fluctuations in Stirring/Exclusion processes with boundary-induced particle creation and annihilation. The methods developed extend the mathematical understanding of non-conservative stochastic particle systems, supplying sharp asymptotics and rigorous convergence to infinite-dimensional Ornstein-Uhlenbeck limits with nontrivial boundary contributions. The careful $v$ -function analysis established here sets a new benchmark for further investigation into non-equilibrium phenomena in interacting particle systems with complex boundary dynamics.