For paths $x:[0, T] \rightarrow \mathbb{R}^n$, the Stratonovich integral along a one form $\omega$ on $\mathbb{R}^n$ can be defined by $$ S_\omega(x) := \int_0^T \omega(x(t)) \mathrm{d}x(t) := \lim_{|\tau|\rightarrow 0} \int_0^T \omega(x^\tau(t)) \mathrm{d}x^\tau(t),$$ where the limit is taken over any sequence of partitions $\tau = \{0 = \tau_0 < \tau_1 <\dots < \tau_N = T\}$ of the interval $[0, T]$ the mesh of which tends to zero and $$x^\tau(\tau_{j-1} + t) := x(\tau_{j-1}) + \frac{t}{\tau_j-\tau_{j-1}} (x(\tau_j)-x(\tau_{j-1}))$$ is the corresponding polygon-approximation of the path $x$.
Clearly, the right hand sides of the above definition are continuous (even smooth) functions on the space $W:=C([0, T], \mathbb{R}^n)$ of continuous paths, but it is well known that limit only exists in probability with respect to the Wiener measure on $W$, and the Stratonovich integral $S_\omega$ ends up being only a function in $L^1(W)$ instead of in $C(W)$.
Now the question is if this can be fixed somehow: Is there a topological space $W^\prime$ of continuous paths with $$\bigcap_{\alpha < 1/2} C^\alpha([0, T], \mathbb{R}^n) \subseteq W^\prime \subseteq W$$ such that the Wiener measure can be constructed as a Borel probability measure on $W^\prime$ and such that $S_\omega \in C(W^\prime)$? More precisely, is there such a space $W^\prime$ so that the net of functions $$S_\omega^\tau(x) := \int_0^T \omega(x^\tau(t)) \mathrm{d}x^\tau(t) = \int_0^T \omega(x^\tau(t)) \dot{x}^\tau(t)\mathrm{d}t$$ converges to a continuous function $S_\omega$? Can $S_\omega$ even be a smooth map?
In other words: Can the solution map constructed from the theory of rough paths be made continuous/smooth when the path space carries the right topology?
