کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
976706 933148 2007 8 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات فیزیک ریاضی
پیش نمایش صفحه اول مقاله
A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations” by T.D. Frank
چکیده انگلیسی
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 [T.D. Frank, Stochastic feedback, non-linear families of Markov processes, and nonlinear Fokker-Planck equations, Physica A 331 (2004) 391]. Our comment centers on the claims of a “non-linear Markov process” and a “non-linear Fokker-Planck equation.” First, memory in transition densities is misidentified as a Markov process. Second, the paper assumes that one can derive a Fokker-Planck equation from a Chapman-Kolmogorov equation, but no proof was offered that a Chapman-Kolmogorov equation exists for the memory-dependent processes considered. A “non-linear Markov process” is claimed on the basis of a non-linear 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 conditional probabilities (the transition probabilities), is either an ordinary linearly generated Markovian one, or else is a linearly generated non-Markovian 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 “non-linear Markov process”, nor a “non-linear Fokker-Planck equation” for a conditional probability density. The confusion rampant in the literature arises in part from labeling a non-linear diffusion equation for a 1-point probability density as “non-linear Fokker-Planck,” whereas neither a 1-point density nor an equation of motion for a 1-point density can define a stochastic process. In a closely related context, we point out that Borland misidentified a translation invariant 1-point probability density derived from a non-linear diffusion equation as a conditional probability density. Finally, in the Appendix A we present the theory of Fokker-Planck pdes and Chapman-Kolmogorov equations for stochastic processes with finite memory.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Physica A: Statistical Mechanics and its Applications - Volume 382, Issue 2, 15 August 2007, Pages 445-452
نویسندگان
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