= Support of enhanced Brownian motion
{title2=$\operatorname{supp}\mathcal L(\mathbf B)=\overline{\{S_2(h):h\in H_0^1\}}^{d_p}$}
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>.
Back to article page
