Stochastic analysis for the Dirichlet--Ferguson process

TL;DR

This paper develops a Malliavin calculus framework for the Dirichlet--Ferguson process, including chaos expansion, operators, and applications to Fleming--Viot process analysis.

Contribution

It introduces a new Malliavin calculus tailored for the Dirichlet--Ferguson process, with explicit formulas and applications to stochastic process generators.

Findings

Reproved chaos expansion with explicit kernel formulas.

Developed a Malliavin calculus with gradient, divergence, and generator.

Identified the generator as that of the Fleming--Viot process and described the Dirichlet form.

Abstract

We study a Dirichlet--Ferguson process $\zeta$ on a general phase space. First we reprove the chaos expansion from Peccati (2008), providing an explicit formula for the kernel functions. Then we proceed with developing a Malliavin calculus for $\zeta$. To this end we introduce a gradient, a divergence and a generator which act as linear operators on random variables or random fields and which are linked by some basic formulas such as integration by parts. While this calculus is strongly motivated by Malliavin calculus for isonormal Gaussian processes and the general Poisson process, the strong dependence properties of $\zeta$ require considerably more combinatorial efforts. We apply our theory to identify our generator as the generator of the Fleming--Viot process and to describe the associated Dirichlet form explicitly in terms of the chaos expansion. We also establish the product and chain chain rule for the gradient and an integral representation of the divergence. Finally we give a short direct proof of the Poincar\'e inequality.