Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank

Details Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank Markov vs. nonMarkovian processes A comment on the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T.D. Frank

2007-01-24

arxiv.org/abs/cond-mat/0701589v5

Physics

Condensed Matter

Statistical Mechanics

Scientific paper

10.1016/j.physa.2007.03.020

The purpose of this comment is to correct mistaken assumptions and claims made in the paper Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations by T. D. Frank. Our comment centers on the claims of a nonlinear Markov process and a nonlinear Fokker-Planck equation. First, memory in transition densities is misidentified as a Markov process. Second, Frank assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was given that a Chapman-Kolmogorov equation exists for memory-dependent processes. A nonlinear Markov process is claimed on the basis of a nonlinear diffusion pde for a 1-point probability density. We show that, regardless of which initial value problem one may solve for the 1-point density, the resulting stochastic process, defined necessarily by the transition probabilities, is either an ordinary linearly generated Markovian one, or else is a linearly generated nonMarkovian process with memory. We provide explicit examples of diffusion coefficients that reflect both the Markovian and the memory-dependent cases. So there is neither a nonlinear Markov process nor nonlinear Fokker-Planck equation for a transition density. The confusion rampant in the literature arises in part from labeling a nonlinear diffusion equation for a 1-point probability density as nonlinear Fokker-Planck, whereas neither a 1-point density nor an equation of motion for a 1-point density defines a stochastic process, and Borland misidentified a translation invariant 1-point density derived from a nonlinear diffusion equation as a conditional probability density. In the Appendix we derive Fokker-Planck pdes and Chapman-Kolmogorov eqns. for stochastic processes with finite memory.

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