The Support of the Law of a Filter in C ∞ Topology

In their well known paper, [18], D.W. Stroock and S. Varadhan proved that the support of the law of a diffusion process solution of a stochastic differential equation: $$ d{x_t} = {X_0}({x_t})dt + {X_i}({x_t})^\circ dw_t^i$$ can be described as the closure (for the natural Banach topology on C([0,1], ℝn) of the set of solutions of the following controlled systems: $$\begin{array}{*{20}{c}} {\dot{x}_{t}^{u} = {{X}_{0}}\left( {x_{t}^{u}} \right) + {{X}_{i}}\left( {x_{t}^{u}} \right)\dot{u}_{t}^{i}, u \in {{H}^{1}}\left( {\left[ {0,1} \right],{{\mathbb{R}}^{n}}} \right)} \hfill \\ {x_{o}^{u} = {{x}_{0}}} \hfill \\ \end{array} $$