Stochastic analysis for the Dirichlet--Ferguson process

We study a Dirichlet–Ferguson process $ζ$ 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 $ζ$. 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 $ζ$ 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 Poincaré inequality.

💡 Research Summary

The paper develops a Malliavin calculus for the Dirichlet–Ferguson (DF) process, a random probability measure whose finite‑dimensional marginals are Dirichlet distributions. After recalling the basic construction of the DF process on a measurable space (X,𝔛) with a finite directing measure ρ (total mass θ), the authors re‑derive the chaos expansion originally proved by Peccati (2008). They show that any square‑integrable functional F of the DF process can be written uniquely as

 F = 𝔼