Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces

Abstract

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.