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Phase space density and fluid flow: Conservation laws and speculations on a Boltzmann equation associated with the stochastic Newton equation. (English) Zbl 0835.60102

Albeverio, S. (ed.) et al., Stochastic processes, physics and geometry II. Proceedings of the 3rd international conference held in Locarno, Switzerland, 24-29 June 1991. Singapore: World Scientific. 491-502 (1995).
Summary: The stochastic Newton equation, in either its fluid form or expressed in terms of conditional expectations with respect to the underlying diffusion, is not directly amenable to an explicit computation of solutions. We propose a Boltzmann-type equation in the hope that methods of the classical kinetic theory may be brought to bear on this problem. The equation is motivated through formal mathematical and physical arguments. In particular, we review how the fluid form of the stochastic Newton equation may be derived as a conservation equation in both the Lagrangian and Eulerian descriptions of the “fluid”, where the former description is intimately connected with certain infinite-dimensional geometries. We conclude by suggesting that there is a natural stochastic process on phase space related to this Boltzmann equation, and to the fluid form of the stochastic Newton equation in the hyperbolic scaling limit.
For the entire collection see [Zbl 0843.00042].

MSC:

60K40 Other physical applications of random processes
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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