Stroock-Varadhan support theorem ($\mathrm{supp}\mathcal{L}(Y)=\overline{\{y^h:h\in H_0^1\}}$)

For smooth bounded vector fields, the support of a Stratonovich diffusion in uniform path topology is the closure of solutions to the controlled ordinary differential equation with the same drift and controls in the Cameron-Martin space of Wiener measure. The support of enhanced Brownian motion and the universal limit theorem prove this through continuity of the enhanced solution map. For an equation written in Itô form, the controlled drift is the Itô drift minus $\frac{1}{2}{\sum }_{i}D{V}_i{V}_i$.