Non-Markovian effects on the Brownian motion of a free particle

Non-Markovian effects on the Brownian movement of a free particle in the presence as well as in the absence of inertial force are investigated under the framework of generalized Fokker–Planck equations (Rayleigh and Smoluchowski equations). More specifically, it is predicted that non-Markovian features can diminish the values of both the root mean square displacement and the root mean square momentum, thereby assuring the mathematical property of analyticity of such physically observable quantities for all times t≥0. Accordingly, the physical concept of non-Markovian Brownian trajectory turns out to be mathematically well defined by differentiable functions for all t≥0. Another consequence of the non-Markovicity property is that the Langevin stochastic equations underlying the Fokker–Planck equations should be interpreted as genuine differential equations and not as integral equations according to a determined interpretation rule (Doob’s rule, for instance).