# Functional inequalities for marked point processes

## TL;DR

This paper extends functional inequalities from Poisson measures to marked temporal point processes, providing new inequalities and representations that have broad applications in stochastic process analysis.

## Contribution

It introduces new functional inequalities for marked point processes, including Poincaré and transportation cost inequalities, extending known results to more general processes.

## Key findings

- Derived a Poincaré inequality for marked point processes.
- Proved two new transportation cost inequalities.
- Applied results to renewal, nonlinear Hawkes, and Cox processes.

## Abstract

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincar\'e inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes.