Propagation and generation of chaos is an important ingredient for rigorous control of applicability of kinetic theory, in general. Chaos is here understood as sufficient statistical independence of random variables related to the "kinetic" observables of the system. Cumulant hierarchy of these random variables thus often gives a way of controlling the evolution and degree of such independence, i.e., the degree of chaos in the system. In this talk, I will discuss our analysis of the cumulant hierarchy of the stochastic Kac model in the preprint [arxiv.org:2407.17068], a joint work with Aleksis Vuoksenmaa. We control generation of chaos via the magnitude of finite order cumulants of kinetic energies for arbitrary symmetric initial data, with the usual restriction of a fixed energy density. This allows estimating the accuracy of kinetic theory, uniformly in time and for any system with sufficiently large number of particles, $N$. We prove that the evolution of the system can be divided into three regimes: an initial regime of the length of at most $O(\ln N)$ in which the state of the system becomes chaotic, the kinetic regime which is determined by solutions to the Boltzmann-Kac equation, and an equilibrium regime. I will also discuss preliminary results for a modification of the model in which the collision rates increase with faster velocity. This appears to lead to shorter, order one, generation of chaos times.

Dynamical systems are inevitably subject to noise perturbations, making the stability of invariant measures under noise perturbations a fundamental problem. Such a stability is well-known for physical measures in hyperbolic systems, but remains widely open for more general systems. This talk will present some recent results on stochastic stability of physical measures in both conservative and dissipative systems.

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.