Module Free White Noise Flows

摘要

The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Ito tables, both classical and quantum and, combined with the Accardi-Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Ito algebra. We prove that any such derivation is the difference of a star-homomorphism and a trivial embedding.