Random walks associated with nonlinear Fokker-Planck equations

Abstract

A nonlinear random walk related to the porous medium equation (nonlinear Fokker-Planck equation) is investigated. This random walk is such that when the number of steps is sufficiently large, the probability of finding the walker in a certain position after taking a determined number of steps approximates to a q-Gaussian distribution (Gq,β(x) ∝ [1 - (1 - q)βx2]1/(1-q)), which is a solution of the porous medium equation. This can be seen as a verification of a generalized central limit theorem where the attractor is a q-Gaussian distribution, reducing to the Gaussian one when the linearity is recovered (q → 1). In addition, motivated by this random walk, a nonlinear Markov chain is suggested.