We consider the one-dimensional stirring process on the segment $\{−N , . . . , N \}$, coupled to boundary dynamics that inject particles from the right reservoir and remove particles from the left reservoir, each acting on a window of fixed and finite size. In this talk, I will present the non-equilibrium fluctuations of the system when the initial configuration is given by a product measure associated with a smooth macroscopic profile. In this regime, the fluctuations are described by an Ornstein–Uhlenbeck process driven by the Laplacian and gradient operators, with boundary conditions determined by the hydrodynamic profile. A central step in the analysis is the derivation of sharp bounds for space and space–time v-functions of arbitrary degree associated with 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 $N^{ −1−\zeta}$ for some $\zeta> 0$.

Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenaria. In particular, alternating patterns between synchronization and desynchronization behaviors are given by studying the asymptotics of the Markov perturbed stationary distributions. This talk is based on joint works with Arno Berger, Wen Huang, Hong Qian, Felix X.-F. Ye, and Yingfei Yi.

A spectral gap bound is an essential ingredient in the rigorous derivation of the macroscopic equations from microscopic stochastic dynamics. This work establishes a spectral gap of order $\ell^{-2}$ for a special class of the simple symmetric mass migration misanthrope process on a finite box $\Lambda_\ell$ with reflecting boundaries. The model allows for cooperative jumps of blocks of any size up to the origin site occupancy, with misanthrope rates governed by the occupation numbers of the departure and arrival sites. By imposing and subsequently exploiting a linear structure in the expected total particle jump rates, using results from the theory of Ollivier-Ricci curvature and classical results on symmetric random walks, we extend the framework of Gobron and Saada [GS10] to accommodate these large-block dynamics.

[GS10] T. Gobron and E. Saada, Couplings, attractiveness and hydrodynamics for conservative particle systems, Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1132–1177.