A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations
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-dimensio...
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| Format: | UnknownFormat |
| Sprache: | eng |
| Veröffentlicht: |
Berlin
WIAS
2004
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| Schriftenreihe: | Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik
994 |
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Tag hinzufügen | Zusammenfassung: | 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. |
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| Beschreibung: | 14 S. graph. Darst. |