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

TL;DR

This paper extends known convergence rate results for ergodic stochastic differential equations driven by fractional Brownian motion to cases with multiplicative noise, using novel Foster-Lyapunov techniques in a non-Markovian context.

Contribution

It introduces a new approach to analyze convergence rates for fractional SDEs with multiplicative noise, extending previous results to more complex noise structures.

Findings

Convergence rates of order t^{-eta} are established for certain fractional SDEs.

Foster-Lyapunov techniques are adapted to non-Markovian settings.

Asymptotic coupling schemes are successfully implemented without deterministic contraction.

Abstract

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\textgreater{}1/2$ 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 $H\textgreater{}1/2$ 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.