The Jacobi Field of a Lévy Process

Abstract

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