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A dimensionreduced description of general Brownian motion by nonautonomous diffusionlike 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–Plancklike 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 diffusionlike 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 (nonGaussian) noise. Using a representation with pseudodifferential operators, we derive a reduced diffusionlike equation, which turns out to be nonautonomous 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 diffusionlike equation), and the free particle case.
Key words and phrases:
Fokker–Planck equation, general Brownian motion, dimensionreduction, pseudodifferential operator.
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MSC: 60J65, 47G10, 47G30, 35S30, 82C31, 35C15 Received: 26.09.2004
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Citation:
Holger Stephan, “A dimensionreduced description of general Brownian motion by nonautonomous diffusionlike equations”, Mat. Fiz. Anal. Geom., 12:2 (2005), 187–202
Citation in format AMSBIB
\Bibitem{Ste05}
\by Holger Stephan
\paper A dimensionreduced description of general Brownian motion by nonautonomous diffusionlike equations
\jour Mat. Fiz. Anal. Geom.
\yr 2005
\vol 12
\issue 2
\pages 187202
\mathnet{http://mi.mathnet.ru/jmag182}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2152645}
\zmath{https://zbmath.org/?q=an:1073.60081}
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Stephan H., “Modeling of driftdiffusion systems”, Z. Angew. Math. Phys., 60:1 (2009), 33–53

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