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Matematicheskaya Fizika, Analiz, Geometriya [Mathematical Physics, Analysis, Geometry], 2005, Volume 12, Number 2, Pages 187–202
(Mi jmag182)
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This article is cited in 2 scientific papers (total in 2 papers)
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
Abstract:
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.
Key words and phrases:
Fokker–Planck equation, general Brownian motion, dimension-reduction, pseudodifferential operator.
Received: 26.09.2004
Citation:
Holger Stephan, “A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations”, Mat. Fiz. Anal. Geom., 12:2 (2005), 187–202
Linking options:
https://www.mathnet.ru/eng/jmag182 https://www.mathnet.ru/eng/jmag/v12/i2/p187
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