New results on Fokker–Planck equations of fractional order

Abstract

It is shown that the fractional Fokker–Planck equations proposed recently in the literature (by merely substituting time fractional derivative for time derivative) give rise to some problems in the sense that they provide probability densities which may have negative values. In the same way, one shows that the Kramers–Moyal equation can be thought of as related to fractal processes, but it is well known that it yields also negative densities. It seems that the key of this trouble is the misuse of the Chapman Kolmogorov equation on the one hand, and of the fractional difference on the other hand. In fact, there is a complete identification between Kramers–Moyal equation and Fokker–Planck equation of fractional order. After a careful analysis, one arrives at the conclusion that the fractional derivative in Liouville–Riemann (L–R) sense should be replaced by a slightly finite fractional derivative which involves finite difference, whilst L–R fractional derivative refers to difference of infinite order. The new fractional Fokker–Planck equation so obtained is displayed, and its solution via separation of variables is outlined. It seems that there is no alternative but to work via non-standard analysis, that is to say infinitesimal discretization in time.