Upper bounds on Rubinstein distances on configuration spaces and applications

TL;DR

This paper establishes upper bounds on Rubinstein distances on Poisson configuration spaces using Malliavin gradients, with applications to process comparison, tail bounds, and isoperimetric inequalities.

Contribution

It introduces new upper bounds on Rubinstein distances involving Malliavin gradients, tailored to different configuration distances, and applies these to process comparison and tail estimates.

Findings

Derived effective bounds for Rubinstein distances on Poisson spaces

Identified which gradient yields the best bounds depending on the configuration distance

Applied bounds to compare Poisson with other processes and to tail and isoperimetric estimates

Abstract

In this paper, we provide upper bounds on several Rubinstein-type distances on the configuration space equipped with the Poisson measure. Our inequalities involve the two well-known gradients, in the sense of Malliavin calculus, which can be defined on this space. Actually, we show that depending on the distance between configurations which is considered, it is one gradient or the other which is the most effective. Some applications to distance estimates between Poisson and other more sophisticated processes are also provided, and an application of our results to tail and isoperimetric estimates completes this work.