R\'egularisation de l'\'equation de Langevin en dimension 1 par le mouvement Brownien fractionnaire

TL;DR

This paper introduces a fractional Langevin equation driven by fractional Brownian motion to model complex physical systems with long memory effects, extending the classical Langevin framework.

Contribution

It proposes a novel fractional stochastic differential equation based on fractional Brownian motion, capturing long-range dependencies in 1D physical phenomena.

Findings

Generalization of Langevin equation to include fractional Brownian motion

Provides a framework for modeling systems with memory effects

Extends classical Langevin dynamics to fractional context

Abstract

The main goal of this paper is to provide a fractional stochastic differential equation modelling the physical phenomena governed by the Langevin equation in 1-dimension. A generalized equation leaning on the fractional Brownian motion (fBm) will be proposed, the later will allow a description of the complexity of the physical systems which escape any prediction of the of the standard Langevin equation. We shall begin at first to remind the basic notions of the standard Brownian motion (Bm) and the fractional Brownian motion (fBm), then, we shall establish a generalization to long memory of the Langevin equation.