Nonsingular transformations of the symmetric L\'evy processes

A. M. Vershik ; N. V. Smorodina

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 102-114

Résumé

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 $\alpha2$ 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.