Nonsingular Transformations of Symmetric Lévy Processes

In this paper, we consider a group of transformations of the space of trajectories of symmetric α-stable Lévy laws with stability constant α ∈ [0; 2). For α = 0, the true analog of a stable Lévy process (the so-called 0-stable process) is the gamma process, whose law is quasi-invariant under the action of the group of multiplicators $\mathcal{M}\equiv \left\{ {{M_a}:a\geq 0,\,\log a\in {L^1}} \right\}$; the action of M_a on a trajectory ω(∙) is given by (M_aω)(t) = a(t)ω(t). For every α < 2, an appropriate conjugacy transformation sends the group $\mathcal{M}$ to the group ${{\mathcal{M}}_a}$ of nonlinear transformations of trajectories, and the law of the corresponding stable process is quasi-invariant under this group. We prove that for α = 2, an appropriate change of coordinates reduces the group of symmetries to the Cameron-Martin group of nonsingular translations of trajectories of the Wiener process.