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Zap. Nauchn. Sem. POMI, 2012, Volume 408, Pages 102–114 (Mi znsl5495)  

This article is cited in 1 scientific paper (total in 2 paper)

Nonsingular transformations of the symmetric Lévy processes

A. M. Vershika, N. V. Smorodinab

a St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia

Abstract: 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.

Key words and phrases: Wiener measure, gamma-mesure, deformation of the symmery groups.

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English version:
Journal of Mathematical Sciences (New York), 2014, 199:2, 123–129

Bibliographic databases:

UDC: 519.2
Received: 08.10.2012

Citation: A. M. Vershik, N. V. Smorodina, “Nonsingular transformations of the symmetric Lévy processes”, Probability and statistics. Part 18, Zap. Nauchn. Sem. POMI, 408, POMI, St. Petersburg, 2012, 102–114; J. Math. Sci. (N. Y.), 199:2 (2014), 123–129

Citation in format AMSBIB
\Bibitem{VerSmo12}
\by A.~M.~Vershik, N.~V.~Smorodina
\paper Nonsingular transformations of the symmetric L\'evy processes
\inbook Probability and statistics. Part~18
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 408
\pages 102--114
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5495}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3032211}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2014
\vol 199
\issue 2
\pages 123--129
\crossref{https://doi.org/10.1007/s10958-014-1839-6}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84902244754}


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    2. I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Probabilistic Approximation of the Evolution Operator”, Funct. Anal. Appl., 52:2 (2018), 101–112  mathnet  crossref  crossref  isi  elib
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