Abstract
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.
Highlights
The stochastic processes studied by McKean satisfy drift-diffusion equations for probability densities that are nonlinear with respect to their probability densities
Nonlinear stochastic processes evolving on discrete state spaces but exhibiting a continuous time variable have been studied in the context of nonlinear master equations as introduced in Section 2.4; that is, in the context of Markov processes
Within the framework of strongly nonlinear stochastic processes this implies that the leverage ratio dynamics of companies depends on an expectation value of the leverage ratio process
Summary
With these notations at hand, some benchmark examples of strongly nonlinear stochastic processes can be given. For processes that evolve on discrete state spaces and satisfy a nonlinear master equation it follows that (35) becomes the iterative map (22) provided that the argument P(x, t) in f(⋅) is expressed in terms of the occupation probability p(k, t); see (6). For processes evolving on continuous state spaces and satisfying strongly nonlinear Ito-Langevin equations, the general form (35) reduces to the stochastic difference equation (26) when we consider the limiting case ξ → 0. Model (37) has been examined in physics, social sciences, and psychology as will be discussed
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