Field Theory for Point Processes

Random Measures, Empirical Fields, and Variational Structure
Erscheinungsjahr 2027
ISBN: 978-3-032-42887-5
Verlag: Birkhäuser

Buch, Englisch, Format (B × H): 155 mm x 235 mm

Reihe: Progress in Probability

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.

Zielgruppe

Research

Autoren/Hrsg.

Clark, Daniel

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik
• Naturwissenschaften Physik Angewandte Physik Statistische Physik, Dynamische Systeme

Inhaltsverzeichnis

Point Processes and Random Measures.- Random Fields and Generalized Functions.- Functional Calculus, Generating Functionals, and Convex Analysis.- Empirical Fields and Functional Representations of Point Processes.- Scale, Renormalization, and Coarse Observation.- Large Deviations and Effective Action.- Stochastic Population Fields and Configuration Operators.- Lagrangians, Hamiltonians, and Actions for Population Fields.- Path Integrals, Large Deviations, and Fluctuation Theory.- Information, Estimation, and Precision Geometry.-Deformation, Symmetry, and Precision Constraints.- Detection, Decision, and Variational Tests.

Über Autor(innen)

Professor Daniel Clark is Chair in Electronics and Computer Science at the University of Southampton, UK, where he is Head of the Vision, Learning, and Control Research Group. He previously served as Head of the Research Group “Traitement de l’Information pour Images et Communications” at Télécom SudParis, Institut Polytechnique de Paris (2020–2022). His research lies at the interface of probability theory, stochastic modeling, and statistical inference, with particular emphasis on point processes, random measures, and stochastic population models. A central theme of his work is the development of functional and variational methods for complex stochastic systems, drawing on large deviation theory, statistical mechanics, and field-theoretic ideas. Professor Clark received his Ph.D. from Heriot-Watt University and holds an Habilitation à Diriger des Recherches (HdR) from Université Paris-Saclay.