A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems—As a reply to McCauley's 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 McCauley's 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.