In this talk we present recent developments concerning a general scheme that has been used to study the stochastic dynamics of a continuous interacting particle system, whether of the birth-and-death type or of the hopping particles type.

We prove that a parabolically rescaled and suitably renormalised height function of a weakly asymmetric simple exclusion process on a circle converges to the Cole-Hopf solution of the KPZ equation. This is an analogue of the celebrated result by Bertini and Giacomin from 1997 for the exclusion process on a circle with any particles density. The main goal of this article is to analyse the interacting particle system using the framework of regularity structures without applying the Gärtner transformation, a discrete version of the Cole-Hopf transformation which linearises the KPZ equation. Our analysis relies on discretisation framework for regularity structures developed by Erhard and Hairer [AIHP 2019] as well as estimates for iterated integrals with respect to jump martingales derived by Grazieschi, Matetski and Weber [PTRF 2025]. The main technical challenge addressed in this work is the renormalisation procedure which requires a subtle analysis of regularity preserving discrete convolution operators. Joint work with R. Huang (Münster) and K. Matetski (Michigan State).

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.

In this talk we consider two interacting particle systems in the finite box with N sites, namely, independent random walks and symmetric exclusion processes. Both systems are in contact with a finite reservoir, where the exit rate is proportional to the inverse of $N^\theta$, where $theta$ is nonnegative. We prove the hydrodynamic limit for these models, which are given by the heat equation with Wentzell boundary conditions at the boundary at the critical parameter $\theta=1$, and exhibit a dynamic phase transition. Moreover, the Wentzell boundary condition is non-linear in the exclusion setting at the critical parameter. Joint work with Patrícia Gonçalves and Matheus Franco.

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.