Support of enhanced Brownian motion ($\mathrm{supp}\mathcal{L}(\mathbf{B})={\overline{\{S_2(h):h\in H_0^1\}}}^{d_p}$)

For $2<p<3$, the support of a measure of enhanced Brownian motion is the rough-path closure of path signatures of paths in the Cameron-Martin space of Wiener measure. Polygonal convergence gives one inclusion. For the reverse inclusion, decompose Brownian motion into its polygonal interpolation and independent pinned bridges. The bridge enhancement becomes small in probability as the mesh decreases, while every finite-dimensional neighborhood of the polygonal vertices has positive Gaussian probability. Rough path translation then gives positive probability in every ball around a prescribed control path signature.