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NUMERIC AND EXACT SOLUTIONS OF THE NONLINEAR CHAPMAN–KOLMOGOROV EQUATION: A CASE STUDY FOR A NONLINEAR SEMI-GROUP MARKOV MODEL

T. D. FRANK

Department of Psychology, Center for the Ecological Study of Perception and Action, University of Connecticut, 406 Babbidge Road, Storrs, CT 06269, USA

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https://doi.org/10.1142/S0217979209053497Cited by:0 (Source: Crossref)

Abstract

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.

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