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Let $H_0^1$ be the Cameron-Martin space of Wiener measure on $[0,T]$: its elements start at zero, are absolutely continuous, and have square-integrable derivatives. For $h\in H_0^1$, let $y^h$ solve the controlled ordinary differential equationThe Stroock-Varadhan support theorem in the uniform topology isOne may equivalently take piecewise linear or smooth controls starting at zero. The skeleton uses the displayed vector fields, as appropriate for the Stratonovich equation.

Choose $2<p<3$ and work in the complete separable space of anchored geometric p-rough paths, with its rough path metric $d_p$. The enhanced Brownian motion is a random element of this space. Its support of a measure isWe explain this Brownian support step using the permitted results about the enhancement. The almost-sure polygonal convergence proved in Question 3 gives the inclusion of its support in $K$. Indeed, polygonal controls belong to $H_0^1$, and the enhancement belongs to their closed signature closure almost surely. Hölder rough-path convergence at an exponent greater than $1/p$ implies $p$-variation convergence: for the level-$k$ differences, sum the bound proportional to $|t-s{|}^{kp\alpha /k}=|t-s{|}^{p\alpha }$ over a partition. Since $p\alpha >1$, these sums are uniformly bounded and tend to zero with the Hölder distance.

For the reverse inclusion, use Brownian rough path small-ball positivity, namely $\mathbb{P}(d_p(\mathbf{B},e)<r)>0$ for every $r>0$, and Cameron-Martin quasi-invariance of enhanced Brownian motion. These are the Brownian input results used without proof here. For $h\in H_0^1$, the rough path translation $T_h$ is continuous and sends $e$ to $S_2(h)$. The law of $T_h\mathbf{B}$ is equivalent to that of $\mathbf{B}$. Given an open neighborhood $U$ of $S_2(h)$, continuity supplies an identity neighborhood mapped into $U$. Small-ball positivity gives positive probability that $T_h\mathbf{B}\in U$; equivalence of the two laws gives $\mathbb{P}(\mathbf{B}\in U)>0$. Hence $S_2(h)$ belongs to the Brownian support for every $h$. Closedness of support proves the equality with $K$.

Let $\Phi$ be the state-path solution map of the rough differential equation driven by these vector fields, with the fixed initial value $y_0$. The assumed identification of the Stratonovich solution says $Y=\Phi (\mathbf{B})$. Boundedness of the fields and all their derivatives ensures global existence and the regularity needed by the universal limit theorem. That theorem makes $\Phi$ continuous from the rough-path space into the uniform state-path space. For a control signature, consistency with ordinary equations gives $\Phi (S_2(h))=y^h$.

We use the elementary support under a continuous map identityTo prove it, a neighborhood of $f(x)$ with $x\in \mathrm{supp}\mu$ has an open inverse image containing $x$, and therefore positive measure. This puts $f(\mathrm{supp}\mu )$, and its closure, in the image support. Conversely, the image closure is closed and has full image measure, since the original support has full measure. The minimal closed full-measure characterization from Question 4(i) gives the other inclusion.

Apply this identity to $f=\Phi$ and the Brownian rough-path law. We obtainContinuity and the definition of $K$ imply $\Phi (K)\subset \overline{\{\Phi (S_2(h)):h\in H_0^1\}}$. The reverse inclusion after taking closures follows from $S_2(h)\in K$. Therefore the last displayed support is exactly the uniform closure of the skeleton paths, proving the theorem.

Finally, piecewise constant and smooth approximations of $\dot{h}$ converge in $L^2$, hence in $L^1$ on a finite interval. Their controls converge in total variation, their signatures converge in the rough-path metric, and the universal limit theorem gives uniform convergence of the corresponding skeleton solutions. This proves the stated equivalence of the control classes.

In the truncated tensor algebra, let ${\mathfrak{g}}^N(V)$ be the linear span of all iterated Lie brackets of elements of $V$ of length at most $N$, with bracket $[a,b]=a\otimes b-b\otimes a$. Brackets of length greater than $N$ vanish. The free step-N nilpotent Lie group isThe exponential and logarithm are the finite seriesIts multiplication is the tensor multiplication above, its identity is $e=(1,0,\dots ,0)$, and its inverse isThe truncated Baker--Campbell--Hausdorff formula shows that this multiplication stays in the exponential of the Lie algebra. This is the connected simply connected Lie group of the free step-$N$ nilpotent Lie algebra, rather than a discrete free nilpotent group.

Give the first layer $V$ its usual Euclidean norm. An absolutely continuous curve $\gamma$ in $G^N(V)$ is horizontal iffor an absolutely continuous control $x$ in $V$. In the truncated tensor algebra this equation is ${\dot{\gamma }}_t={\gamma }_t\otimes {\dot{x}}_t$, truncated at degree $N$. Its horizontal length is $\int |{\dot{x}}_t|dt$. The Carnot-Carathéodory distance isThe first-layer Lie brackets span the full Lie algebra, so horizontal curves reach every endpoint. Thus this infimum is finite; it is a left-invariant distance inducing the usual manifold topology. Dilations ${\delta }_{\lambda }(a{)}^{(k)}={\lambda }^k{a}^{(k)}$ give $d_{CC}({\delta }_{\lambda }g,{\delta }_{\lambda }h)=\lambda d_{CC}(g,h)$ for $\lambda >0$. Equivalently $d_{CC}(e,g)$ is the least length of a bounded variation control whose step-$N$ path signature is $g$.

For $p\ge 1$ set $N=\lfloor p\rfloor$. A weak geometric p-rough path is a continuous path $\mathbf{X}:[0,T]\to G^N(V)$, usually based at $e$, withIts increments ${\mathbf{X}}_{s,t}={\mathbf{X}}_s^{-1}\otimes {\mathbf{X}}_t$ obey ${\mathbf{X}}_{s,u}={\mathbf{X}}_{s,t}\otimes {\mathbf{X}}_{t,u}$. Equivalently their degree-$k$ tensor components have finite $p/k$-variation and satisfy the path signature algebraic constraints. This is a group-valued definition; being a limit of smooth path signatures in the same $p$-variation topology is the stronger definition of a geometric p-rough path.