The correlated dichotomous noise as an exact M-Gaussian stochastic process

Gaussian statistics is a milestone and often a fundamental assumption in the treatment of complex phenomena. The relevance of Gaussian statistics is highlighted by the Central Limit Theorem (CLT), which states that the sum of a large number of independent random variables leads to a universal form of the probability density function (PDF). A proposal for a sound extension of the definition of “Gaussianicity” and, consequently, of the CLT (named M-CLT) was presented in a recent paper [M. Bianucci, Chaos, Sol. & Frac. 148 (2021) 110961]. This extension includes non commuting quantities (named q-numbers or operators) and can be done in different ways, by choosing different projection mappings (the M map) to be combined with the exponentials entering in the definitions of both the first and the second characteristic functions. In the same paper it was illustrated that there is a strict relationship between “M“-Gaussianicity and the generalized Fokker Planck Equation (FPE). Here we show that there exists a particular M-map for which the dichotomous noise is M-Gaussian and the vector field of the system of interest perturbed by such a noise is M-Gaussian as well. Thus, for such a system of interest, a generalized FPE, characterized by a memory kernel, can be obtained. This is an interesting result in itself, but it is also important for two practical reasons: first, according to the M-CLT, it follows that such time-non local FPE is the M-Gaussian limit expression to which converge, under some assumptions, the sum (or the integral) of stochastic first order differential operators. Second, it states that, being M-Gaussian, the dichotomous noise assumes a relevance, in the frame of statistical processes, comparable to that of the standard Gaussian ones.