TY - JOUR

T1 - Nonsingular Transformations of Symmetric Lévy Processes

AU - Vershik, A.M.

AU - Smorodina, N.V.

PY - 2014

Y1 - 2014

N2 - 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 M≡{Ma:a≥0,loga∈L1}; the action of Ma on a trajectory ω(∙) is given by (Maω)(t) = a(t)ω(t). For every α <2, an appropriate conjugacy transformation sends the group M to the group Ma 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.

AB - 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 M≡{Ma:a≥0,loga∈L1}; the action of Ma on a trajectory ω(∙) is given by (Maω)(t) = a(t)ω(t). For every α <2, an appropriate conjugacy transformation sends the group M to the group Ma 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.

U2 - 10.1007/s10958-014-1839-6

DO - 10.1007/s10958-014-1839-6

M3 - Article

VL - 199

SP - 123

EP - 129

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -