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Stochastic analysis for the Dirichlet--Ferguson process

Günter Last, Babette Picker

Abstract

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.

Stochastic analysis for the Dirichlet--Ferguson process

Abstract

We study a Dirichlet--Ferguson process 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 . 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 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.
Paper Structure (14 sections, 43 theorems, 290 equations)

This paper contains 14 sections, 43 theorems, 290 equations.

Table of Contents

  1. Introduction
  2. The Dirichlet--Ferguson process
  3. Chaos expansion
  4. Malliavin operators
  5. The gradient
  6. The divergence
  7. The Fleming--Viot operator
  8. The Fleming--Viot process
  9. Further properties of the Malliavin operators
  10. Covariance identities
  11. The Poincaré inequality
  12. Orthogonality relations
  13. Integral formulas
  14. A summation formula for rising factorials

Key Result

Theorem 3.3

Suppose that $F\in L^2(\zeta)$. Then the chaos expansion EqChaosZerlegung holds with the functions $f_n$, $n\in{\mathbb N}$, given by kerneln. The functions $f_n$ are $\rho^{[n]}$-a.e. uniquely determined by EqChaosZerlegung.

Theorems & Definitions (97)

  • Theorem 3.3
  • Remark 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • proof
  • Lemma 3.8
  • proof
  • ...and 87 more