This is a question that bothered me for a long time. For more than in one occasion I thought I found a satisfactory answer just to realize after few months I was wrong. Consider a compact Riemannian manifold $M$. We will consider the coordinate process $X$ on the path space of $M$. Now in section $8.5$ of Elton P. Hsu's book Stochastic Analysis on Manifolds it is mentioned that the anti-development of $X$ and $X$ generate the same filtration after completions without any source or argument (I've seen this statement in some other introductory documents on this subject too). Is there any source on this result or is it just a folklore result? My second question is that, can we say the same about general diffusions on path spaces ?i.e. diffusions and their anti-developments generate the same filtration.
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