Unitary representation of walks along random vector fields and the Kolmogorov–Fokker–Planck equation in a Hilbert space

Random Hamiltonian flows in an infinite-dimensional phase space is represented by random unitary groups in a Hilbert space. For this, the phase space is equipped with a measure that is invariant under a group of symplectomorphisms. The obtained representation of random flows allows applying the Chernoff averaging technique to random processes with values in the group of nonlinear operators. The properties of random unitary groups and the limit distribution for their compositions are described.