In the first part of the talk I recall the stochastic version of the Koopman
program and its quantum extension. In the second part I outline an algebraic approach to
dynamical systems with emphasis on the role of 1-cocycles and multiplicative functionals. Then
I describe the structure of cocycles and multiplicative functionals that are not differentiable but
only stochastically differentiable. The third part of the talk goes in some sense in the converse
direction of the first part.