An approach via fractional analysis to non-linearity induced by coarse-graining in space

Loosely speaking, a coarse-grained space is a space in which the generic point is not infinitely thin, but rather has a thickness; and here this feature is modelled as a space in which the generic increment is not dx, but rather (dx)α, 0<α<10<α<1. The purpose of the article is to analyse the non-linearity induced by this coarse-graining effect. This approach via (dx)α leads us to the use of fractional analysis which thus provides models in the form of nonlinear differential equations of fractional order. Two illustrative examples are considered. In the first one, one shows that a particle which has a Gaussian white noise in a coarse-grained spaces exhibits a trajectory which looks like a generalised fractional Brownian motion. The second example shows how a simple one-dimensional linear dynamics is converted into a non-linear system of fractional order.