Intertwining and the Markov uniqueness problem on path spaces

In `Stochastic partial differential equations and application--VII', 89-95, Lect. Notes Pure Appl. Math., 245, Chapman & Hall/CRC, Boca Raton, FL, 2006 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.