Conflict between trajectories and density description: the statistical source of disagreement

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We study an idealized version of intermittent process leading the fluctuations of a stochastic dichotomous variable ξ. It consists of an overdamped and symmetric potential well with a cusp-like minimum. The right-hand and left-hand portions of the potential corresponds to ξ= W and ξ= -W, respectively. When the particle reaches this minimum is injected back to a different and randomly chosen position, still within the potential well. We build up the corresponding Frobenius-Perron equation and we evaluate the correlation function of the stochastic variable ξ, called Φξ(t). We assign to the potential well a form yielding Φξ(t) = (T/(t + T))β, with β> 0. We limit ourselves to considering correlation functions with an even number of times, indicated for concision, by <12>, <1234> and, more, in general, by <1... 2n>. The adoption of a treatment based on density yields <1... 2n > = < 1 2 >... < (2n-1) 2n>. We study the same dynamic problem using trajectories, and we establish that the resulting two-time correlation function coincides with that afforded by the density picture, as it should. We then study the four-times correlation function and we prove that in the non-Poisson case it departs from the density prescription, namely, from <1234 > = < 12 > < 34>. We conclude that this is the main reason why the two pictures yield two different diffusion processes, as noticed in an earlier work [M. Bologna, P. Grigolini, B.J. West, Chem. Phys. 284, (1-2) 115-128 (2002)].