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.
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