Fock Space Decomposition of Levy Processes

Abstract

We show that the general Lévy process can be embedded in a suitable Fock space, classified by cocycles of the real line regarded as a group, $\mathbf{R}$. The formula of de Finetti corresponds to coboundaries. Kolmogorov's processes correspond to cocycles of which the derivatives are cocycles of the Lie algebra of $\mathbf{R}$. Lévy's formula gives the most general cocycle possible.