%0 Journal Article
%T Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences
%A T. D. Frank
%J ISRN Mathematical Physics
%D 2013
%R 10.1155/2013/149169
%X Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics, engineering sciences, biology, psychology, social sciences, finance, and economics. 1. Introduction 1.1. McKean and a New Class of Stochastic Processes In two seminal papers, McKean introduced a new type of stochastic processes [1, 2]. The stochastic processes studied by McKean satisfy drift-diffusion equations for probability densities that are nonlinear with respect to their probability densities. While nonlinear drift-diffusion equations have frequently been used to describe how concentration fields evolve in time [3], the models considered by McKean describe stochastic processes. A stochastic process not only describes how expectation values (such as the mean value) evolve in time, but also, provides a complete description of all possible correlations between two or more than two time points [4¨C6]. Consequently, the study of stochastic processes goes beyond the study of concentration fields or the evolution of single-time point expectation values such as the mean. The observation by McKean that nonlinear drift-diffusion equations can describe stochastic processes opened a new avenue of research. 1.2. From McKeanĄŻs Stochastic Processes to Strongly Nonlinear
%U http://www.hindawi.com/journals/isrn.mathematical.physics/2013/149169/