Summary: A one-dimensional Fokker-Planck equation with nonmonotonic exponentially dependent drift and diffusion coefficients is defined by further generalizing a previously studied “unifying stochastic Markov process.” The equation, which has six essential parameters, defines and unifies a large class of interdisciplinary relevant stochastic processes, many of them being “embedded” as limiting cases. In addition to several known processes that previously have been solved independently, the equation also covers a wide “interpolating” variety of different, more general stochastic systems that are characterized by a more complex state dependence of the stochastic forces determining the process. The systems can be driven by additive and/or multiplicative noises. They can have saturating or nonsaturating characteristics and they can have unimodal or bimodal equilibrium distributions. Mathematically, the generalization considered parallels the extension from the Gauss hypergeometric to the Heun differential equation, by adding one more finite regular singularity and its associated confluence possibilities. A previously developed constructive solution method, based upon double integral transforms and contour integral representation, is extended for the actual equation by introducing “factorizers” and by using a few of their fundamental properties (compiled in Appendix A).
In addition, the equivalent Schrödinger equation and the reflection symmetry principle are proved to be important tools for analysis. Fully analytical results including normalization are obtained for the discrete part of the generally mixed spectrum. Only the eigenvalues have to be numerically determined as zeros of a spectral kernel. This kernel generally is unknown, but its zeros are accessible via appropriate, infinite continued fraction based search schemes. The basic role of “congruence” in this context is highlighted. For clarity, the simpler standard case corresponding 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 the principles for deriving the limiting results in confluent cases with fully discrete spectra. Possible fields of application are, e.g., population dynamics in biology, noise in nonlinear electronic circuits, chemical and nuclear reaction kinetics, systems with noise-induced transitions or transitions to bimodality, genetics, and neural network stochastics.