November 2023 On the (non)stationary density of fractional-driven stochastic differential equations
Xue-Mei Li, Fabien Panloup, Julian Sieber
Author Affiliations +
Ann. Probab. 51(6): 2056-2085 (November 2023). DOI: 10.1214/23-AOP1638


We investigate the stationary measure π of SDEs driven by additive fractional noise with any Hurst parameter and establish that π admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener–Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the nonstationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this, we revisit the fractional averaging principle of Li and Sieber (Ann. Appl. Probab. 32 (2022) 3964–4003) and remove an ad hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements.

Funding Statement

XML has been supported by the EPSRC Grants EP/V026100/1 and EP/S023925/1; JS has been supported by G-Research and the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).


We thank the three anonymous referees for their helpful comments. Xue-Mei Li is also affiliated with École Polytechnique Fédérale de Lausanne, would like to acknowledge support from EPFL.


Download Citation

Xue-Mei Li. Fabien Panloup. Julian Sieber. "On the (non)stationary density of fractional-driven stochastic differential equations." Ann. Probab. 51 (6) 2056 - 2085, November 2023.


Received: 1 April 2022; Revised: 1 January 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1638

Primary: 37A25 , 60G22 , 60H10

Keywords: fractional Brownian motion , Gaussian-type bounds , Girsanov theorem , parameter-dependent SDE , smooth density , smoothness of the invariant density , Wiener–Liouville bridge

Rights: Copyright © 2023 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.51 • No. 6 • November 2023
Back to Top