A dimension-reduced description of general Brownian motion by non-autonomous diffusion-like equations
- Stephan, Holger
2010 Mathematics Subject Classification
- 60J65 47G10 47G30 35S30 82C31 35C15
- Fokker-Planck equation, general Brownian motion, dimension-reduction, pseudodifferential operator
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.
- J. Mathematical Physics Analysis Geometry (MAG), 12 (2005) pp. 187-202.