  There are two open problem on the analysis of continuous paths on a
Riemannian manifold, the Markov uniqueness and the independence of the closure
of the differential operator $d$ on its initial domain. The operator $d$ acts
naturally on $BC^1$ functions, one is concerned with its extensions to the
$L^2$ spaces. With a suitable choice of an initial domain we denote by
$D^{2,1}$ its closure under the graph norm. For the Wiener space, the domain of
$d$ can be classified, as a consequence its extension is unique whether the
initial domain is smooth cylindrical or in $BC^1$ etc. This has not shown to be
the same when the measure is the probability distribution of any smooth
elliptic diffusion. In an earlier paper, we have shown that the closure of
$BC^\infty$ functions agree with that of smooth cylindrical functions, leaving
an undesirable gap. The Markov uniqueness is essentially concerned with the
problem whether there exists a unique Markov process on the path space whose
Markov generator agrees with the infinite-dimensional Laplacian on $C^\infty$
cylindrical functions. Here we reduce Markov uniqueness to whether the pull
back of $D^{2,1}$ by the ito map is $ D^{2,1}$ (i.e. a surjection). We also
propose a possible approach for tackle this problem.
