Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise

Joaquin Fontbona (1) , Fabien Panloup (2)

1 Departamento de Ingenieria Matematica, Centro Modelamiento Matematico
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in \cite{hairer} that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha\in(0,1)$ (depending on $H$). The aim of this paper is to extend such types of results to some multiplicative noise setting. More precisely, we show that we can recover such convergence rates when $H>1/2$ and the inverse of the diffusion coefficient $\sigma$ is a Jacobian matrix. The main novelty of this work is a type of extension of Foster-Lyapunov like techniques to this non-Markovian setting, which allows us to put in place an asymptotic coupling scheme such as in \cite{hairer} without resorting to deterministic contracting properties.

Mots clés: Fractional Brownian Motion; Stochastic Differential Equations; Total variation distance; Ergodicity; Multiplicative noise; Rate of convergence to equilibrium; Lyapunov function; Total variation distance.

Domaines: Probabilités [math.PR]

Joaquin Fontbona, Fabien Panloup. Rate of convergence to equilibrium of fractional driven stochastic differential equations with some multiplicative noise. Annales de l'Institut Henri Poincaré, 2017, 53 (2), ⟨10.1214/15-AIHP724⟩. ⟨hal-00989414v2⟩