Long time Hurst regularity of fractional SDEs and their ergodic means

The fractional Brownian motion can be considered as a Gaussian field indexed by $(t,H)\in {\mathbb{R}}_{+}\times (0,1)$, where $H$ is the Hurst parameter. On compact time intervals, it is known to be almost surely jointly Hölder continuous in time and Lipschitz continuous in $H$. First, we extend this result to the whole time interval ${\mathbb{R}}_{+}$ and consider both simple and rectangular increments. Then we consider SDEs driven by fractional Brownian motion with contractive drift. The solutions and their ergodic means are proven to be almost surely Hölder continuous in $H$, uniformly in time. This result is used in a separate work for statistical applications. We also deduce a sensibility result of the invariant measure in $H$. The proofs are based on variance estimates of the increments of the fractional Brownian motion and fractional Ornstein-Uhlenbeck processes, multiparameter versions of the Garsia-Rodemich-Rumsey lemma and a combinatorial argument to estimate the expectation of a product of Gaussian variables.