Variable separation without preselection and non-spectral relaxation in Fokker-Planck and generalized Fokker-Planck equations

点击翻译按钮获取中文摘要

The relaxation of a dissipative system to its equilibrium state often shows a multiexponential pattern with relaxation rates, which are typically considered to be independent of the initial condition. The rates follow from the spectrum of a Hermitian operator obtained by a similarity transformation of the initial Fokker-Planck operator. However, some initial conditions are mapped by this similarity transformation to functions which grow at infinity. These cannot be expanded in terms of the eigenfunctions of a Hermitian operator, and show different relaxation patterns. Such situations are exotic for Gaussian noises but are the rule in the Levy case. Being left without such a powerful tool of theoretical analysis of such systems as spectral decomposition, we have to look for alternatives.