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Математическая физика, анализ, геометрия, 2005, том 12, номер 2, страницы 187–202
(Mi jmag182)
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Эта публикация цитируется в 2 научных статьях (всего в 2 статьях)
A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations
Holger Stephan Weierstrass Institute for Applied Analysis and Stochastics, 39 Mohrenstrasse, 10117 Berlin, Germany
Аннотация:
The Brownian motion of a classical particle can be described by a Fokker–Planck-like equation. Its solution is a probability density in phase space. By integrating this density w.r.t. the velocity, we get the spatial distribution or concentration. We reduce the $2n$-dimensional problem to an $n$-dimensional diffusion-like equation in a rigorous way, i.e., without further assumptions in the case of general Brownian motion, when the particle is forced by linear friction and homogeneous random (non-Gaussian) noise. Using a representation with pseudodifferential operators, we derive a reduced diffusion-like equation, which turns out to be non-autonomous and can become elliptic for long times and hyperbolic for short times, although the original problem was time homogeneous. Moreover, we consider some examples: the classical Brownian motion (Gaussian noise), the Cauchy noise case (which leads to an autonomous diffusion-like equation), and the free particle case.
Ключевые слова и фразы:
Fokker–Planck equation, general Brownian motion, dimension-reduction, pseudodifferential operator.
Поступила в редакцию: 26.09.2004
Образец цитирования:
Holger Stephan, “A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations”, Матем. физ., анал., геом., 12:2 (2005), 187–202
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/jmag182 https://www.mathnet.ru/rus/jmag/v12/i2/p187
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