NUMERIC AND EXACT SOLUTIONS OF THE NONLINEARCHAPMAN-KOLMOGOROV EQUATION: A CASE STUDYFOR A NONLINEAR SEMI-GROUP MARKOV MODEL

摘要

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