$\varepsilon > 0$ by Kolmogorov’s continuity theorem, and thus have $\frac{1}{p}$-Hölder regularity for any $p > 2\rho = \frac{1}{H}$. The proof that it has finite 2D $\rho$-variation can be found in [@fv2010a]; see also [@fv2011].

In the case $1 \leq p < 2$, or $H > \frac{1}{2}$, the geometric rough path is simply $\left( 1, B^H_t \right)$, and for $H \leq \frac{1}{2}$, one can invoke Theorem \[gaussianRP\] to construct the geometric rough path. Now Condition \[newCond1\] is satisfied since we have $\mathcal{H}^d \hookrightarrow \mathcal{C}^{q-var} \left([0, T]; \mathbb{R}^d\right)$ from Proposition 17 in [@fv2010a] and Theorem 1 in [@fggr2016]. Note that the second case, which applies when $\frac{1}{4} < H \leq \frac{1}{3}$, follows from the fact that the covariance function has finite mixed $(1, \rho)$-variation (cf. [@fggr2016]).

For any $H \in (0, 1)$, we have (see Theorem 3.2 in [@du1999]) $$\begin{aligned}
 \label{kBound1}
|K (t,s)| \leq C_{1, H} s^{-\left| H - \frac{1}{2} \right|} (t - s)^{-\left( \frac{1}{2} - H \right)},\end{aligned}$$ for all $0 < s < t \leq T$, and we also have $$\begin{aligned}
 \label{kBound2}
\frac{\partial K(t, s)}{\partial t} = C_{2, H} \left( \frac{t}{s} \right)^{H - \frac{1}{2}} (t - s)^{- \left(\frac{3}{2} - H \right)};\end{aligned}$$ see [@ccm2003] and [@nualart2006]. Thus, Condition \[amnCond\] is satisfied 