Geodesic
Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations
Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 697-712

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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.
Keywords: Poisson measure, spectral measure, gamma measure, Hellinger–Kakutani distance.
Mots-clés : quasi-invariance 
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     title = {Poisson {Measures} {Quasi-Invariant} with {Respect} to {Multiplicative} {Transformations}},
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M. A. Lifshits; E. Yu. Shmileva. Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations. Teoriâ veroâtnostej i ee primeneniâ, Tome 46 (2001) no. 4, pp. 697-712. http://geodesic.mathdoc.fr/item/TVP_2001_46_4_a4/