N.Tsilevich, A.Vershik, M.Yor

Quasi-invariance of the gamma process and multiplicative properties
of the Poisson--Dirichlet measures.

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 (the fact first
discovered by Gelfand, Graev and Vershik) and the Markov--Krein identity as
corollaries of the formula for the Laplace transform of $P_\Ga$.

The quasi-invariance of the measure $P_\Ga$ allows us to obtain new
quasi-invariance properties of the measure $PD(\theta)$. The corresponding
invariance properties hold for $\si$-finite analogues of $P_\Ga$ and $PD(\th)$.
We also show that the measure $P_\Gamma$ can be considered as a limit of
measures corresponding to the $\al$-stable L\'evy 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.