\; \bigg\vert \; t\in[0,T],
\; u = 1, \ldots, d \right\}\end{aligned}$$ under the inner-product $$\begin{aligned}
\left\langle R(t, \cdot)^{(u)}, R(s,\cdot)^{(v)} \right\rangle _{\mathcal{H}^{d}} = \delta_{uv} \, R(t,s), \quad u, v = 1, \ldots, d.\end{aligned}$$ By definition, $\mathcal{H}^{d}$ satisfies the following reproducing property; for any $f=\left(  f^{(1)},\ldots, f^{(d)}\right)  \in\mathcal{H}^{d}$, $$\begin{aligned}
\left\langle f_{\cdot}, R(t,\cdot)^{(u)} \right\rangle_{\mathcal{H}^{d}} 
= f_{t}^{(u)}, \quad t \in[0,T].\end{aligned}$$

We assume that there exists $\rho < 2$ such that $R$ has finite 2D $\rho$-variation. The following theorem in [@fv2010a] (see also [@cq2002] in the case of fractional Brownian motion) then shows that one can canonically lift $X$ via its piecewise linear approximants $X^{\pi}$ to a geometric $p$-rough path for $p > 2 \rho$.

\[gaussianRP\] Assume $X$ is a centered continuous $\mathbb{R}^d$-valued Gaussian process with i.i.d. components. Let $\rho \in [1, 2)$ and assume that the covariance function has finite 2D $\rho$-variation.

(i) (Existence) There exists a random variable $\mathbf{X}=\left( 1, \mathbf{X}^{1}, \mathbf{X}^2, \mathbf{X}^3\right) $ on $\left( \Omega, \mathcal{F},\mathbb{P}\right) $ which takes values in $\mathcal{C}^{0, p-var} \left( [0, T]; G^3(\mathbb{R}^d) \right)$ for $p > 2\rho$ almost surely, and is hence a geometric $p-$rough path for $p \in (2\rho, 4)$. Moreover, $\mathbf{X}$ lifts the Gaussian process $X$ in the sense that $\mathbf{X}_{s,t}^{1}=X_{t}-X_{s}$ almost surely for 