The Poisson Multiplication Formula

TL;DR

This paper provides necessary and sufficient conditions for the square-integrability of products of Poisson functionals, introduces novel Poincaré inequalities, and generalizes multiplication formulas on the Poisson space, addressing longstanding open questions.

Contribution

It establishes new conditions for Poisson functional products, introduces a novel family of Poincaré inequalities, and generalizes multiplication formulas on the Poisson space.

Findings

Conditions for square-integrability of Poisson functional products

New Poincaré inequalities for finite random variables

General multiplication formulas under minimal conditions

Abstract

We establish necessary and sufficient conditions implying that the product of $m\geq 2$ Poisson functionals, living in a finite sum of Wiener chaoses, is square-integrable. Our conditions are expressed in terms of iterated add-one cost operators, and are obtained through the use of a novel family of Poincar\'e inequalities for almost surely finite random variables, generalizing the recent findings by Trauthwein (2024). When specialized to the case of multiple Wiener-It\^o integrals, our results yield general multiplication formulae on the Poisson space under minimal conditions, naturally expressed in terms of partitions and diagrams. Our work addresses several questions left open in a seminal work by Surgailis (1984), and completes a line of research initiated in D\"obler and Peccati (2018).