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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?

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  • $\begingroup$ I'm fuzzy on the details, but since you mention rough paths, isn't this precisely what the $p$-variation topology accomplishes (with $p > 2$)? $\endgroup$ – Nate Eldredge Aug 11 '16 at 21:54
  • $\begingroup$ Is this so? I am not an expert on the theory of rough paths and tried to find such a result in the literature, but I could not find the statement. Maybe this is just a language barrier, or maybe this way of thinking about matters is uncommon for probabilists. $\endgroup$ – Matthias Ludewig Aug 12 '16 at 7:06
  • $\begingroup$ The result I'm thinking of is that the rough path integral is supposed to be a continuous map on the space of enhanced rough paths (paths in the truncated tensor algebra) when it's equipped with the appropriate $p$-variation topology. But that's not what's going on here. Anyway, the enhancement that leads to the Stratonovich integral is something like Stratonovich stochastic area, and it is only defined almost everywhere. $\endgroup$ – Nate Eldredge Aug 12 '16 at 8:07
  • $\begingroup$ I meant the intersection instead of the union, sorry. However, you where saying that this solution map is continuous, and then again, that it is only defined almost everywhere? $\endgroup$ – Matthias Ludewig Aug 12 '16 at 8:42
  • $\begingroup$ The map from the space of enhanced rough paths to the integral is continuous. The map from the space of actual paths to enhanced rough paths is only a.e. defined. What you want is their composition. $\endgroup$ – Nate Eldredge Aug 12 '16 at 8:46

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