  In this paper we consider the Koopman operator associated with the discrete
and the continuous time random dynamical system (RDS). We provide results that
characterize the spectrum and the eigenfunctions of the stochastic Koopman
operator associated with different types of linear RDS. Then we consider the
RDS for which the associated Koopman operator family is a semigroup, especially
those for which the generator can be determined. We define a stochastic Hankel
DMD (sHankel-DMD) algorithm for numerical approximations of the spectral
objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator and
prove its convergence. We apply the methodology to a variety of examples,
revealing objects in spectral expansions of the stochastic Koopman operator and
enabling model reduction.
