RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Zh. Mat. Fiz. Anal. Geom.: Year: Volume: Issue: Page: Find

 Mat. Fiz. Anal. Geom., 2005, Volume 12, Number 2, Pages 187–202 (Mi jmag182)

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.

Full text: PDF file (283 kB)
Full text: http:/.../abstract.php?uid=m12-0187e

Bibliographic databases:
MSC: 60J65, 47G10, 47G30, 35S30, 82C31, 35C15
Language:

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

Citation in format AMSBIB
\Bibitem{Ste05} \by Holger Stephan \paper A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations \jour Mat. Fiz. Anal. Geom. \yr 2005 \vol 12 \issue 2 \pages 187--202 \mathnet{http://mi.mathnet.ru/jmag182} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2152645} \zmath{https://zbmath.org/?q=an:1073.60081} 

• http://mi.mathnet.ru/eng/jmag182
• http://mi.mathnet.ru/eng/jmag/v12/i2/p187

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Stephan H., “Modeling of drift-diffusion systems”, Z. Angew. Math. Phys., 60:1 (2009), 33–53
•  Number of views: This page: 517 Full text: 27