Unitary representation of walks along random vector fields and the Kolmogorov-Fokker-Planck equation in a Hilbert space. (English. Russian original) Zbl 1536.37070
Theor. Math. Phys. 218, No. 2, 205-221 (2024); translation from Teor. Mat. Fiz. 218, No. 2, 238-257 (2024).
Summary: 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.
MSC:
| 37K65 | Hamiltonian systems on groups of diffeomorphisms and on manifolds of mappings and metrics |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 47H40 | Random nonlinear operators |
| 35Q84 | Fokker-Planck equations |
| 58J65 | Diffusion processes and stochastic analysis on manifolds |
Keywords:
random operator; random Hamiltonian flow; invariant measure; Weil theorem; Gaussian random walk; Laplace-Volterra operator; Sobolev space; Kolmogorov-Fokker-Planck equationReferences:
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