Chapter

Continuous Markov process theory

Daniel T. Gillespie and Effrosyni Seitaridou

in Simple Brownian Diffusion

Published in print October 2012 | ISBN: 9780199664504
Published online January 2013 | e-ISBN: 9780191748516 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199664504.003.0007
Continuous Markov process theory

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Three years after Einstein published his groundbreaking 1905 paper on diffusion, a different analysis was published by Paul Langevin. Although it was not clearly appreciated at the time, Langevin's approach avoided the shortcomings of Einstein's approach that were described in Chapter 4, yet included Einstein's theory as a special limiting approximation. A full understanding of all this requires the mathematical machinery of continuous Markov process theory, which is essentially a generalization to the stochastic realm of the deterministic theory of ordinary differential equations. This chapter is a self-contained exposition of those topics in continuous Markov process theory that are needed to understand Langevin's model of diffusion and how it is related to Einstein's model. The topics covered include: the Chapman-Kolmogorov and Kramers-Moyal equations; the process increment; the self-consistency requirement on the process increment, and how that requirement gives rise to the Langevin equation;(n) Gaussian white noise; the forward and backward Fokker-Planck equations; multivariate continuous Markov processes; the driftless Weiner process, and how it can be numerically simulated; and the Ornstein-Uhlenbeck process and its time-integral, and how they can be numerically simulated.

Keywords: Chapman-Kolmogorov equation; Kramers-Moyal equation; process increment; self-consistency condition; Langevin equation; stochastic differential equation; drift function; diffusion function; Gaussian white noise; Weiner process; Ornstein-Uhlenbeck process

Chapter.  14714 words. 

Subjects: Physics

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