Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations

In this work the necessary and sufficient conditions are given for the quasi-invariance of the distributions of Poisson measures on $X\times\mathbf{R}^+$ (for arbitrary measurable space $X$) with respect to a large group of the scalings of the component $\mathbf{R}^+$. It is shown that the class of quasi-invariant measures is far from being exhausted by the measures absolutely continuous with respect to the gamma measure considered in [N. Tsilevich and A. Vershik, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), pp. 163–168] and [N. Tsilevich, A. Vershik, and M. Yor, Prepublication 575, Universites Paris VI & Paris VII, Paris, 2000]. A criterion is given for the absolute continuity of a Poisson measure with respect to another Poisson measure on an arbitrary measurable space.