  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 H\"older
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 H\"older continuous in $H$, uniformly in time. This result
is used in a separate work for statistical applications. 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.
