Nonextensive It\^o-Langevin Dynamics

Abstract

We study generalizations of It\^{o}-Langevin dynamics consistent within nonextensive thermostatistics. The corresponding stochastic differential equations are shown to be connected with a wide class of nonlinear Fokker-Planck equations describing correlated anomalous diffusion in fractals. A generalized central limit theorem is proposed in order to demonstrate how such equations emerge as a limit of correlated random variables. In doing so, we connect microscopic and macroscopic descriptions of correlated anomalous diffusion in a mathematically sound way and shed some light in explaining why $q$-Gaussian distributions appear quite often in nature.