Nonlinear Fokker-Planck equation: stability, distance and corresponding extremal problem in the spatially inhomogeneous case
arXiv:1307.1126 · doi:10.1007/978-3-319-10335-8
Abstract
We start with a global Maxwellian , which is a stationary solution, with the constant total density ($ρ(t)\equiv \wt ρ$), of the Fokker-Planck equation. The notion of distance between the function and an arbitrary solution (with the same total density $\wt ρ$ at the fixed moment ) of the Fokker-Planck equation is introduced. In this way, we essentially generalize the important Kullback-Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.
Minor text corrections are made in the second version. The paper is published in "Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes" Oper. Theory Adv. Appl. 244, 379--394, Birkhauser, 2015