Interacting particle systems through point processes: static structure and dynamics

Kateryna Hlyniana
Theory of Stochastic Processes
Vol.29 (45), no.2, 2025, pp.30-65
This survey develops a second-order viewpoint on point processes and on configuration-valued stochastic dynamics. We treat point processes as random counting measures on configuration space and emphasize the tools that govern first and second order: factorial moment measures, correlation functions, the pair-correlation function, Campbell-Mecke integrals, and conditional intensities. We then review algebraic classes with explicit correlation structure - determinantal, permanental/Cox, and Pfaffian point processes - highlighting how their kernels constrain repulsion or clustering through g(r). The second part turns to interacting particle systems and flows, using two-point functions to compare lattice models (exclusion, voter, contact), continuum birth-death and Glauber dynamics, and the associated BBGKY-type correlation hierarchies. A central case study is one-dimensional coalescing and annihilating systems: at fixed times they form Pfaffian point processes, yielding explicit formulas for ρ_t⁽¹⁾, ρ_t⁽²⁾ and short-range inhibition induced by collision history, and connecting to the Arratia flow. We conclude with open problems on Pfaffian models with controlled attraction, second-order classification of IPS, and multi-type extensions.
DOI: https://doi.org/10.3842/tsp-9506612372-47
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