Pathwise Large deviations for the pure jump $k$-nary interacting particle systems

TL;DR

This paper establishes a pathwise large deviation principle for pure jump $k$-nary interacting particle systems, extending classical models like Boltzmann and Smoluchowski, with novel methods for bounds and rate function analysis.

Contribution

It introduces a new large deviation framework for $k$-nary particle systems, including novel coupling techniques and a simplified analysis of the rate function.

Findings

Proved the large deviation upper bound using process perturbation methods.

Established the lower bound via orthogonal martingale measures and coupling.

Analyzed the rate function and discussed solutions related to gelation phenomena.

Abstract

A pathwise large deviation result is proved for the pure jump models of $k$-nary interacting particle system introduced by Kolokoltsov that generalize classical Boltzmann's collision model, Smoluchovski's coagulation model and many others. The upper bound is obtained by following the standard methods of using a process "perturbed" by a regular function. To show the lower bound, we propose a family of orthogonal martingale measures and prove a coupling for the general perturbations. The rate function is studied based on the idea of L\'eonard with a simplification by considering the conjugation of integral functionals on a subspace of $L^{\infty}$. General "gelling" solutions in the domain of the rate function are also discussed.