Starting from the Radon measure on \(\mathbb R\setminus \{ 0\}\) with an infinite support, the authors define a Lévy type generalized stochastic process on the space of distributions over a non-compact Riemannian manifold. The \(L^2\)-space with respect to the law of this process is isometric to an extended Fock space and that isometry transforms the multiplication operator into the operator Jacobi field [see also \textit{Yu. M. Berezansky}, Integral Equations Oper. Theory 30, 163--190 (1998; Zbl 0896.60019)]. An explicit formula for the latter is given. The gamma, Pascal, and Meixner processes are characterized as the only Lévy processes whose Jacobi fields leave invariant the set of continuous elements of the extended Fock space with finite numbers of non-zero components.