Celebratio Mathematica

N. Tsilevich, A. Vershik, and M. Yor: Quasi-invariance of the gamma process and multiplicative properties of the Poisson–Dirichlet measures. Preprint, 1999. techreport

In this paper we describe new fundamental properties of the law \( P_{\Gamma} \) of the classical gamma process and related properties of the Poisson–Dirichlet measures \( PD(\theta) \). We prove the quasi-invariance of the measure \( P_{\Gamma} \) with respect to an infinite-dimensional multiplicative group and the Markov–Krein identity as corollaries of the formula for the Laplace transform of \( P_{\Gamma} \). The quasi-invariance of the measure \( P_{\Gamma} \) allows us to obtain new quasi-invariance properties of the measure \( PD(\theta) \). The corresponding invariance properties hold for \( \sigma \)-finite analogues of \( P_{\Gamma} \) and \( PD(\theta) \). We also show that the measure \( P_{\Gamma} \) can be considered as a limit of measures corresponding to the \( \alpha \)-stable Lévy processes when parameter \( \alpha \) tends to zero. Our approach is based on simultaneous considering the gamma process (especially its Laplace transform) and its simplicial part–the Poisson–Dirichlet measures.

N. Tsilevich, A. Vershik, and M. Yor: Distinguished properties of the gamma process and related topics. Preprint 575, March 2000. ArXiv math.PR/0005287 techreport

We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasi-invariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson–Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the Poisson–Dirichlet measures.

N. Tsilevich, A. Vershik, and M. Yor: “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process,” J. Funct. Anal. 185 : 1 (September 2001), pp. 274–296. MR 1853759 Zbl 0990.60053 article

We define a one-parameter family \( \mathscr{L}_{\theta} \) of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand–Graev–Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group \( \operatorname{SL}(2,\mathbb{R})^X \).

A. M. Vershik, M. Yor, and N. V. Tsilevich: “On the Markov–Krein identity and the quasi-invariance of the gamma process,” J. Math. Sci., New York 121 : 3 (2004), pp. 2303–2310. English translation of an article published in Representation theory, dynamical systems, combinatorial and algorithmic methods 6 (2001). MR 1879060 article

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Marc Yor

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