Infinite-dimensional equation Liouville with respect to measures.

*(English. Russian original)*Zbl 1210.37046
Dokl. Math. 81, No. 3, 476-480 (2010); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 432, No. 1, 28-32 (2010).

An infinite system of equations with respect to time-dependent finite-dimensional probability distributions is obtained. This system is equivalent to the Liouville equation with respect to functions of a real argument taking values in the space of probability measures. As a consequence, the equations of this type as referred in this paper as Liouville equations with respect to measures. It is observed that in the absence of an interaction, all equations in this system turn out to be independent. Then, it is shown that the probability distribution on the configuration space tends to a product of independent uniform distributions. The system of equations under consideration substantially differs from the Bogoliubov infinite system of equations. It is proven that the solution of the Liouville infinite-dimensional equation with respect to measures is the pointwise weak limit of the measure-valued functions. Also, it is deduced that the Liouville equation with respect to functions is an equation for the first integrals of the Hamilton equation.

Reviewer: Gheorghe Zet (Iaşi)

##### MSC:

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

35Q70 | PDEs in connection with mechanics of particles and systems of particles |

##### Keywords:

Liouville equation; probability distribution; measures; configuration space; Hamilton equation
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\textit{V. V. Kozlov} and \textit{O. G. Smolyanov}, Dokl. Math. 81, No. 3, 476--480 (2010; Zbl 1210.37046); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 432, No. 1, 28--32 (2010)

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