The overdamped generalized Langevin equation with Hermite noise

Abstract

We consider the generalized Langevin equation driven by a fractional non-Gaussian noise, the so-called Hermite process. We prove the existence and the uniqueness of the solution for this equation as well as various properties. An useful link between the solution to the Langevin equation with Hermite noise and a class of self-similar stochastic processes with stationary increments is given in our work. Based on this relationship, we deduce the regularity of the sample paths of the solution and the behavior of its p-variation. Some applications concerning the statistical estimation of the noise intensity parameter or the Euler method for the generalized Langevin equation are also discussed.