Formulas for distributions of mean values of linear functionals with respect to generalized Dirichlet processes, as well as for joint distributions of mean values of several linear functionals with respect to Dirichlet processes are obtained. In a joint work with S. V. Kerov a multidimensional Markov–Krein transform is introduced and studied. New characterizations of the Poisson–Dirichlet measures are obtained. In a series of papers (joint with A. Vershik and M. Yor) invariance properties of gamma processes are used to introduce and study a family of so called multiplicative measures, including an infinite-dimensional analogue of the Lebesgue measure. The developed theory is applied to studying Poisson–Dirichlet measures, stable processes, Markov–Krein identity and to the representation theory of current groups.
Main publications:
N. Tsilevich, A. Vershik, M. Yor. An infinite-dimensional analogue of the Lebesgue measure, and distinguished properties of the gamma process // J. Funct. Anal., v. 185, no. 1, 274–296, 2001.
N. Tsilevich, A. Vershik. Quasi-invariance of the gamma process and multiplicative properties of the Poisson–Dirichlet measures // C. R. Acad. Sci. Paris, v. 329, Ser. I, p. 163–168, 1999.
