Field Theory for Point Processes: Random Measures, Empirical Fields, and Variational Structure - Daniel Clark

This monograph develops a field-theoretic framework for the analysis of point processes, treating them as random measures and generalized fields rather than purely combinatorial objects. By treating point processes in terms of empirical fields, generating functionals, and effective actions, it offers a unified analytical approach to their structure, dynamics, and inference. Standard constructions, such as Poisson processes, stochastic dynamics, and inference procedures, are recast as components of a single analytic structure. This makes explicit structural correspondences that are often implicit: between generating functions and entropy, cumulant hierarchies and interaction, convex duality and large deviations, stochastic evolution and functional differential equations, and Fisher information and precision geometry.

The intended audience includes researchers and advanced postgraduate students in probability theory, stochastic processes, stochastic geometry, mathematical physics, statistical signal processing, and related areas. The presentation is largely self-contained, but assumes familiarity with measure-theoretic probability and basic functional analysis.