This thesis focusses on the study of multi-layer particle systems with an emphasis on the scaling limits of particle systems of this type. Chapter 1 gives an overall introduction to the field of statistical physics and interacting particle systems, and gives a motivation on the study of multi-layer particle systems. In Chapter 2 we introduce the mathematical background required for this thesis in a rigorous manner. The topics discussed include Markov semigroups and generators, path-space convergence, ergodic theory, martingales, couplings, and duality. In Chapter 3 we introduce three types of multi-layer particle systems; the multi-layer exclusion process, the multi-layer inclusion process and the run-and-tumble particle process. We then characterize the ergodic measures with a finite moment condition for these three processes using duality and successful couplings. In Chapter 4 we study the hydrodynamic limit and the stationary fluctuations of the multi-layer run-and-tumble particle process, and use them to infer the same scaling limits for the total density. Furthermore, by an application of Schilder's theorem, we find a large deviation result for the fluctuation field of the total density. In Chapter 5 we establish a large deviation principle for the multi-species stirring process. The method of proof involves studying the hydrodynamic limit of a weakly asymmetric process and a superexponential estimate. In Chapter 6 we return to the multi-layer setting and establish a large deviation principle for the run-and-tumble particle process on two layers with an added mean-field interaction, meaning that the switching between the layers depends on the magnetization of the process. We end with a first step towards an explicit large deviation principle of the total density.

In this paper we consider the multispecies stirring process on the discrete torus. We prove a large deviation principle for the trajectory of the vector of densities of the different species. The technique of proof consists in extending the method of the foundational paper [15] based on the superexponential estimate to the multispecies setting. This requires a careful choice of the corresponding weakly asymmetric dynamics, which is parametrized by fields depending on the various species. We also prove the hydrodynamic limit of this weakly asymmetric dynamics, which is similar to the ABC model in [12, 2]. Using the appropriate asymmetric dynamics, we also obtain that the mobility matrix relating the drift currents to the fields coincides with the covariance matrix of the reversible multinomial distribution, which then further leads to the Einstein relation.

We study the stationary fluctuations of independent run-and-tumble particles. We prove that the joint densities of particles with given internal state converges to an infinite dimensional Ornstein-Uhlenbeck process. We also consider an interacting case, where the particles are subjected to exclusion. We then study the fluctuations of the total density, which is a non-Markovian Gaussian process, and obtain its covariance in closed form. By considering small noise limits of this non-Markovian Gaussian process, we obtain in a concrete example a large deviation rate function containing memory terms.

We consider a class of multi-layer interacting particle systems and characterize the set of ergodic probability measures with finite moments. The main technical tool is duality combined with successful coupling.