The Jacobi field of a L\'evy process

Abstract

We derive an explicit formula for the Jacobi field that is acting in an extended Fock space and corresponds to an ($\R$-valued) L\'evy process on a Riemannian manifold. The support of the measure of jumps in the L\'evy--Khintchine representation for the L\'evy process is supposed to have an infinite number of points. We characterize the gamma, Pascal, and Meixner processes as the only L\'evy processes whose Jacobi field leaves the set of finite continuous elements of the extended Fock space invariant.