Abstract
I will talk about several properties of stationary solutions of fractional SDEs. I will first recall some seminal results by Hairer (2005) on the construction of stationary solutions and associated ergodic results. Then, I will focus on a recent paper with Xue-Mei Li and Julian Sieber where we prove smoothness and Gaussian bounds for the density of the related invariant distribution (under appropriate assumptions) in the additive case. 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 non-stationary density, which extend previously known results in the additive setting. Avoiding any use of Malliavin calculus in our arguments, our results are obtained under minimal regularity requirements.
References
Li, X. M., Panloup, F., & Sieber, J. (2022). On the (Non-) Stationary Density of Fractional-Driven Stochastic Differential Equations. arXiv preprint arXiv:2204.06329.