Geodesic
Nonsingular transformations of the symmetric Lévy processes
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 102-114 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we consider the group of transformations of the space of trajectories of the symmetric $\alpha$-stable Lévy laws with constant of stability $\alpha\in[0;2)$. For $\alpha=0$ the true analog of the stable Lévy process (so called $0$-stable process) is the $\gamma$-process, whose measure is quasi-invariant under the action of the group of multiplicators $\mathcal M\equiv\{M_a\colon a\geq0;\lg a\in L^1\}$ – the action of $M_a$ on trajectories $\omega(.)$ is $(M_a\omega)(t)=a(t)\omega(t)$. For each $\alpha<2$ an appropriate conjugacy takes the group $\mathcal M$ to a group $\mathcal M_\alpha$ of nonlinear transformations of the trajectories and the law of the corresponding stable process is quasi-invariant under those groups. We prove that when $\alpha=2$, the appropriate changing of the coordinates reduces the group of symmetries to the Cameron–Martin group of nonsingular translations of the trajectories of Wiener process. 
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A. M. Vershik; N. V. Smorodina. Nonsingular transformations of the symmetric Lévy processes. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 102-114. http://geodesic.mathdoc.fr/item/ZNSL_2012_408_a6/

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