Unified description of random motions and generalized Fokker-Planck equations

Abstract

This review surveys diverse types of random motion within a unified framework, introducing essential mathematical tools, including special functions and fractional calculus, in a pedagogical manner. We focus on generalized random motions in one-dimensional space described by Fokker-Planck like equations, whose fundamental solutions take a universal and simple form in the Laplace-Fourier domain in terms of a generic function $φ(s)$. By exploring the functional forms of $φ(s)$ consistent with physical constraints, we derive a broad class of stochastic dynamics. This includes Brownian and active motion, simple and anomalous diffusion, and processes featuring stochastic resetting or trapping mechanisms. For each case, we provide the corresponding kinetic equation, explicit solutions (where available), and mean-square displacements alongside their asymptotic behaviors. Finally, we discuss the physical interpretation of these motions through the lens of time-changed processes and subordinators.