Quasi-invariance and integration by parts for determinantal and permanental processes

TL;DR

This paper proves a quasi-invariance property for determinantal and permanental processes, showing their stability under perturbations and deriving an integration by parts formula using Malliavin calculus techniques.

Contribution

It establishes a quasi-invariance result and an integration by parts formula for determinantal and permanental processes, extending their analytical understanding.

Findings

Atoms exhibit mutual attraction or repulsion.

Perturbed processes remain determinantal or permanental.

Provides an integration by parts formula for these processes.

Abstract

Determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent. Their atoms exhibit mutual attraction of repulsion, thus these processes are very far from the uncorrelated situation encountered in Poisson models. We establish a quasi-invariance result : we show that if atoms locations are perturbed along a vector field, the resulting process is still a determinantal (respectively permanental) process, the law of which is absolutely continuous with respect to the original distribution. Based on this formula, following Bismut approach of Malliavin calculus, we then give an integration by parts formula.