## The Annals of Probability

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

#### 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 $H\in(1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in(0,1)$, it was proved in [Ann. Probab. 33 (2005) 703–758] 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 $H$). In [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538], this result has been extended to the multiplicative case when $H>1/2$. In this paper, we obtain these types of results in the rough setting $H\in(1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 503–538] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.

#### Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 464-518.

Dates
Revised: November 2017
First available in Project Euclid: 13 December 2018

https://projecteuclid.org/euclid.aop/1544691626

Digital Object Identifier
doi:10.1214/18-AOP1265

Mathematical Reviews number (MathSciNet)
MR3909974

Zentralblatt MATH identifier
07036342

#### Citation

Deya, Aurélien; Panloup, Fabien; Tindel, Samy. Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise. Ann. Probab. 47 (2019), no. 1, 464--518. doi:10.1214/18-AOP1265. https://projecteuclid.org/euclid.aop/1544691626

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#### Supplemental materials

• Supplement to “Rate of convergence to equilibrium of fractional driven stochastic differential equations with rough multiplicative noise”. Our supplement develops the proofs of several crucial but technical results in our paper. We first prove a Lyapunov-type property for rough differential equations with inward looking drifts. Then we handle rough differential equations involving singular drifts, a type of system which arises when one tries to condition in the highly non-Markovian fractional Brownian motion setting. Next, we show how to lift rough paths involving singularities. Finally, we evaluate some effects of our conditioning procedure on the underlying fractional Brownian motion $X$ in equation (1.3).