We study the Markov property of processes described by generalized Fokker–Planck equations that are nonlinear with respect to probability densities such as mean field Fokker–Planck equations and Fokker–Planck equations related to generalized thermostatistics. We show that their transient...

For stochastic processes defined by nonlinear Fokker–Planck equations a microscopic dynamics is proposed in terms of generalized Langevin equations. These Langevin equations can, for example, be used to model stochastic processes with mean field interactions and random walks related to the...

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker–Planck equations arising from interactions among random walkers. We extend the...

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...

We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that...

Exact time-dependent solutions representing generalized one-dimensional Ornstein–Uhlenbeck processes are derived from nonlinear Fokker–Planck equations and the corresponding diffusion processes are studied. The corresponding systems are related to the Renyi entropy and entropies proposed by...

We consider a generalized Ornstein–Uhlenbeck process described by a nonlinear Fokker–Planck equation related to the one-parametric, nonextensive Gaussian entropy, which is a special case of the two-parametric Sharma–Mittal entropy. We derive the entire hierarchy of distribution functions...

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary...