On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations

TL;DR

This paper analyzes the stationary and non-stationary densities of fractional-driven SDEs, establishing Gaussian bounds and smoothness properties without Malliavin calculus, and revisits the fractional averaging principle.

Contribution

It introduces a novel Wiener-Liouville bridge representation for the stationary density, enabling Gaussian bounds and smoothness results under minimal regularity assumptions.

Findings

Stationary density admits smooth Lebesgue density with Gaussian bounds.

Gaussian bounds extend to non-stationary densities using the new representation.

Smoothness of the stationary density is established jointly in parameters and space.

Abstract

We investigate the stationary measure $\pi$ of SDEs driven by additive fractional noise with any Hurst parameter and establish that $\pi$ 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 non-stationary 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(5) (2022)] 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.