Path space large deviation on a lattice for interacting particle systems

Abstract

We investigate the path space large deviation principle (LDP) for a class of interacting particle systems on a fixed finite discrete torus. As an illustrative example, we use the partial exclusion process with maximal occupation of $s$ particles. We first verify the standard Kurtz-type law of large numbers for the original and tilted processes, thereby identifying the deterministic flows around which the large deviations are formulated. We then establish the path space LDP, with speed $s$, and identify the associated rate function in variational form as the Legendre transform of a nonlinear Hamiltonian. The proof relies on martingale techniques, exponential tightness, and a change-of-measure argument based on exponential tilting, which renders atypical trajectories typical under a modified dynamics. We also explain how the same strategy extends to further interacting particle systems. The inclusion process illustrates the conservative non-compact finite-jump case. The harmonic process is qualitatively different: a single transition may move a variable number of particles, and the limiting Hamiltonian is logarithmic with a non-trivial domain. This places the harmonic process outside the usual finite-reaction framework for density-dependent Markov jump processes. Finally, we discuss continuous-state energy-exchange models, such as the Brownian energy process and an integrable heat-conduction model, to indicate how the same Hamiltonian large-deviation formalism appears beyond discrete-state particle systems.