Stochastic Canonical Flows

There has been some opposition to the use of Markov processes in quantum mechanics based upon the fact that no quantum mechanical regression principle can be applied. It is argued here however that one may just as well conclude that there is no classical mechanical regression principal either. The dissipative component of a canonical stochastic flow is of the form of a double Poisson bracket 1/2{{·, F_α}F_α} while the vector component of the noise is the Hamiltonian vector field {·, Fα}; thus the dissipative term is smoother than the noise coefficient. In contrast, the classical Langevin theory is based upon the Ornstein-Uhlenbeck process which has linear drift and constant noise coefficient. Preservation of the canonical structure is at odds with the approximation procedure by Ornstein-Uhlenbeck processes and this is the case in both classical and quantum situations.