Transfer operators and conditional expectations: the non-commutative case, the case of mu-Brownian motions and white noise space setting

Our focus is the operators of multivariable stochastic calculus, i.e., systems of transfer operators, covariance operators, conditional expectations, stochastic integrals, and the counterpart infinite-dimensional stochastic derivatives. In this paper, we present a new operator algebraic framework which serves to unify the analysis and the interrelations for the operators in question. Our approach uses Rokhlin decompositions, and it applies to both general classes of Gaussian processes, and white noise probability space, in commutative probability, as well as to the analogous operators in the framework of quantum (non-commutative) probability.