Geodesic
Cauchy problem for stochastic Liouville equation with randomly variable Hamiltonian of perturbations in the form of a bounded operator
Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 1, pp. 101-111 
Yu. N. Barabanenkov. Cauchy problem for stochastic Liouville equation with randomly variable Hamiltonian of perturbations in the form of a bounded operator. Teoretičeskaâ i matematičeskaâ fizika, Tome 42 (1980) no. 1, pp. 101-111. http://geodesic.mathdoc.fr/item/TMF_1980_42_1_a10/
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     author = {Yu. N. Barabanenkov},
     title = {Cauchy problem for stochastic {Liouville} equation with randomly variable {Hamiltonian} of~perturbations in~the form of~a~bounded operator},
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A class of stochastic problems is considered, in which the perturbation hamiltoniarn of dynamic system depends on a random function of time and coordinates (“the potential”). It is assumed that the perturbation hamiltonian is a bounded operator for sufficiently regular realisations of the potential. The condition for the random potential to belong to the measurable real Hilbert space with finite measure as well as the property of potential correlations weakening is formulated in terms of cumulant functions. For the class of problems under consideration, the solution of the stochastic Liouville–Neumann equation is constructed and limiting theorem about the validity of basic kinetic equation is proved, which includes the approximation of weak interaction with external system and the approximation of small density.

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