Narrow Results

Recently by Casas et al. a notion of chain of evolution algebras (CEAs) is introduced. This chain is a dynamical system the state of which at each given time is an evolution algebra. It is known 25 distinct classes of chains of two-dimensional evolution algebras. In our previous paper we gave the classification of two-dimensional real evolution algebras. This classification contains seven (pairwise non-isomorphic) such algebras. For each known CEA we study its dynamics to be an element of a given class.

Nonlinear Markov processes are known to arise from diffusion approximations of nonlinear master equations and generalized Boltzmann equations and have been studied in their own merit in the context of nonlinear Fokker–Planck equations. Nonlinear Markov processes can account for collective phenomena, collective oscillations, and collective chaotic behavior. Despite the importance of nonlinear Markov processes for the physics of stochastic processes, relatively little is known about the Chapman–Kolmogorov equation of nonlinear Markov processes. The manuscript derives the exact solution of the Chapman–Kolmogorov equation for a recently proposed deterministic Markov model. Numerical solutions of the Chapman–Kolmogorov equation are used to demonstrate the nonlinear semi-group property of conditional probabilities of nonlinear Markov processes.

The Chapman–Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman–Kolmogorov equation is obtained. From the fractional Chapman–Kolmogorov equation, the Fokker–Planck equation is derived.

It is shown that superpositions of path integrals with arbitrary Hamiltonians and different scaling parameters υ ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing distributions for υ have a specific functional form. Ensuing "smearing" distributions substantially simplify the coupled system of Fokker-Planck equations for smeared and unsmeared conditional probabilities. Simple application in financial models with stochastic volatility is presented.