This article is cited in 1 scientific paper (total in 1 paper)
A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations
Weierstrass Institute for Applied Analysis and Stochastics, 39 Mohrenstrasse, 10117 Berlin, Germany
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.
PDF file (283 kB)
MSC: 60J65, 47G10, 47G30, 35S30, 82C31, 35C15
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
Citation in format AMSBIB
\by Holger Stephan
\paper A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations
\jour Mat. Fiz. Anal. Geom.
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Stephan H., “Modeling of drift-diffusion systems”, Z. Angew. Math. Phys., 60:1 (2009), 33–53
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