A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems—As a reply to McCauleyʹs comment

One goal of this mini-tutorial is to provide an introduction into the theory of measure-valued Markov processes and nonlinear martingales defined by strongly nonlinear Fokker–Planck equations and to discuss the physical relevance of the associated processes. Another goal is to reply to McCauleyʹs comment on T.D. Frank [Physica A 331, 391 (2004)]. The tutorial addresses in detail two approaches found in physics and mathematics. The first approach exploits a mapping between linear and nonlinear Fokker–Planck equations. The second approach exploits martingale theory. Several examples of Markov processes and martingales in quantum mechanical, nonextensive, and self-organizing systems defined by nonlinear Fokker–Planck equations are discussed.