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

摘要

The purpose of this comment is to correct mistaken assumptions and claimsmade in the paper Stochastic feedback, nonlinear families of Markov processes,and nonlinear Fokker-Planck equations by T. D. Frank. Our comment centers onthe claims of a nonlinear Markov process and a nonlinear Fokker-Planckequation. First, memory in transition densities is misidentified as a Markovprocess. Second, Frank assumes that one can derive a Fokker-Planck equationfrom a Chapman-Kolmogorov equation, but no proof was given that aChapman-Kolmogorov equation exists for memory-dependent processes. A nonlinearMarkov process is claimed on the basis of a nonlinear diffusion pde for a1-point probability density. We show that, regardless of which initial valueproblem one may solve for the 1-point density, the resulting stochasticprocess, defined necessarily by the transition probabilities, is either anordinary linearly generated Markovian one, or else is a linearly generatednonMarkovian process with memory. We provide explicit examples of diffusioncoefficients that reflect both the Markovian and the memory-dependent cases. Sothere is neither a nonlinear Markov process nor nonlinear Fokker-Planckequation for a transition density. The confusion rampant in the literaturearises in part from labeling a nonlinear diffusion equation for a 1-pointprobability density as nonlinear Fokker-Planck, whereas neither a 1-pointdensity nor an equation of motion for a 1-point density defines a stochasticprocess, and Borland misidentified a translation invariant 1-point densityderived from a nonlinear diffusion equation as a conditional probabilitydensity. In the Appendix we derive Fokker-Planck pdes and Chapman-Kolmogoroveqns. for stochastic processes with finite memory.