UNIFICATION OF ONE-DIMENSIONAL FOKKER-PLANCK EQUATIONS BEYOND HYPERGEOMETRICS - FACTORIZER SOLUTION METHOD AND EIGENVALUE SCHEMES

A one-dimensional Fokker-Planck equation with nonmonotonic exponential ly dependent drift and diffusion coefficients is defined by further ge neralizing a previously studied ''unifying stochastic Markov process.' ' The equation, which has six essential parameters, defines and unifie s a large class of interdisciplinary relevant stochastic processes, ma ny of them being ''embedded'' as limiting cases. In addition to severa l known processes that previously have been solved independently, the equation also covers a wide ''interpolating'' variety of different, mo re general stochastic systems that are characterized by a more complex state dependence of the stochastic forces determining the process. Th e systems can be driven by additive and/or multiplicative noises. They can have saturating or nonsaturating characteristics and they can hav e unimodal or bimodal equilibrium distributions. Mathematically, the g eneralization considered parallels the extension from the Gauss hyperg eometric to the Heun differential equation, by adding one more finite regular singularity and its associated confluence possibilities. A pre viously developed constructive solution method, based upon double inte gral transforms and contour integral representation, is extended for t he actual equation by introducing ''factorizers'' and by using a few o f their fundamental properties (compiled in Appendix A). In addition, the equivalent Schrodinger equation and the reflection symmetry princi ple prove to be important tools for analysis. Fully analytical results including normalization are obtained for the discrete part of the gen erally mixed spectrum. Only the eigenvalues have to be numerically det ermined as zeros of a spectral kernel. This kernel generally is unknow n, but its zeros are accessible via appropriate, infinite continued fr action based search schemes. The basic role of ''congruence'' in this context is highlighted. For clarity, the simpler standard case corresp onding to directly accessible zeros is elaborated first in sufficient detail and the necessary extensions are gradually introduced afterward . The different types of solutions known to exist for Heun's equation eigenvalue problems are identified and are seen to have a ''unified'' structure as well. A small selection of case studies proves ''downward '' compatibility with the previous hypergeometric case and sketches th e principles for deriving the limiting results in confluent cases with fully discrete spectra. Possible fields of application are, e.g., pop ulation dynamics in biology, noise in nonlinear electronic circuits, c hemical and nuclear reaction kinetics, systems with noise-induced tran sitions or transitions to bimodality, genetics, and neural network sto chastics.