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

TL;DR

This paper critiques claims of nonlinear Markov processes and Fokker-Planck equations, clarifying that memory effects do not produce true nonlinear Markovian dynamics and providing explicit examples and derivations.

Contribution

The paper clarifies misconceptions by demonstrating that memory-dependent processes are not nonlinear Markov processes and provides explicit examples and derivations for processes with finite memory.

Findings

Memory in transition densities is not Markovian.

Nonlinear diffusion equations do not define nonlinear Markov processes.

Explicit examples of diffusion coefficients reflect both Markovian and non-Markovian cases.

Abstract

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.