Transformations of Lévy Processes

Abstract

A Lévy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k respects the counits maps generators on B to generators on C. A tranformation between the corrresponding two Lévy processes is given by forming infinitesimal convolution products. This general result is applied to various situations, e.g., to a *-bialgebra and its associated primitive tensor *-bialgebra (called "generator process") as well as its associated group-like *-bialgebra (called Weyl-*-bialgebra). It follows that a Lévy process on a *-bialgebra can be realized on Bose Fock space as the infinitesimal convolution product of its generator process such that the vacuum vector is cyclic for the L\e'vy process. Moreover, we obtain convolution approximations of the Azéma martingale by the Wiener process and vice versa.