The goal of this paper is to define and study a notion of fractional Brownian motion on a Lie group. We define it as at the solution of a stochastic differential equation driven by a linear fractional Brownian motion. We show that this process has stationary increments and satisfies a local self-similar property. Furthermore the Lie groups for which this self-similar property is global are characterized.
Fabrice Baudoin. Laure Coutin. "Self-similarity and fractional Brownian motion on Lie groups." Electron. J. Probab. 13 1120 - 1139, 2008. https://doi.org/10.1214/EJP.v13-530