Self-similarity and fractional Brownian motions on Lie groups

TL;DR

This paper introduces a fractional Brownian motion on Lie groups via stochastic differential equations, characterizes its self-similarity properties, and establishes foundational results including an integration by parts formula and density existence.

Contribution

It defines fractional Brownian motion on Lie groups, characterizes when self-similarity is global, and proves key analytical properties like an integration by parts formula.

Findings

Fractional Brownian motion on Lie groups has stationary increments.

Self-similarity is characterized for specific Lie groups.

An integration by parts formula and density existence are established.

Abstract

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. Finally, we prove an integration by parts formula on the path group space and deduce the existence of a density.