Stochastic Koopman program, quantum extensions of classical evolutions and a unified approach to classical and quantum dynamical systems

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.