18.11.2023 || Invariant Measures of Flows in a Hilbert Space and Hamiltonian Random Walks

18.11.2023 Invariant Measures of Flows in a Hilbert Space and Hamiltonian Random Walks Докладчик: V. Zh. Sakbaev, Keldysh Institute of Applied Mathematics Russian Academy of Science, Moscow, Russia We analyse unitary representation of Hamiltonian flows in a real separable Hilbert space E endowed with a shift-invariant symplectic form ω. To this aim we study measures on the Hilbert space that are invariant with respect to the group of smooth symplectomorphisms of the space (E, ω) preserwing two-dimensional symplectic subspaces [1]. Random Hamilton function H and corresponding random Hamilton flow ΦH in the phase space E equipped with an invariant measure µ are considered ( [2]). By means of the Koopman unitary representaion of random nollinear flows Φ in the spaces L2(E, µ) we can apply methods of random linear group averaging [3]. The limit theorem for walks along the random Hamiltonian vector field on the symplectic space are obtained in the term of generalised convergence in distributions. Sobolev spaces and spaces of smooth functions are introduced and used to describing of infinitesimal operators of limit processes. References [1] Sakbaev V. Zh. Flows in infinite-dimensional phase space equipped with a finitely additive invariant measure, Mathematics, 11:5, 1161, 49 pp. (2023). [2] Busovikov V.M., Sakbaev V. Zh. Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces, International Journal of Modern Physics A, 37:20/21, 2243018 (2022). [3] Orlov Yu.N., Sakbaev V.Zh., Shmidt E.V. Compositions of Random Processes in a Hilbert Space and Its Limit Distribution, Lobachevskii J. Math., 44:4, 1432-1447 (2023)