Entropy, reversible diffusion processes and Markov uniqueness (Q1915351)

Two new proofs for Markov uniqueness of pre-Dirichlet forms of type \[ {\mathcal E}_\varphi(f,g)=\int_{\mathbb{R}^d}\langle\nabla f,\nabla g\rangle_{\mathbb{R}^d}\varphi^2 dx,\qquad f,g\in C_0^\infty(\mathbb{R}^d), \] where \(\varphi\in H_{\text{loc}}^{1,2}(\mathbb{R}^d)\), are given. One is probabilistic and based (among other things) on results on uniqueness of the corresponding martingale problem. The other is purely analytic and, in fact, more direct and more elementary than all previous ones. Also the questions whether every invariant measure is reversible and whether it is unique, are addressed. Towards the end of the paper there appear, however, to be some statements requiring some clarification. First, it is not realized that for a function \(\psi\in H^{1,2}(\mathbb{R}^d)\) it does not make sense to speak of the interior of \(\{\psi > 0\}\), since this depends on which Borel-version one considers. Second, in contrast to what is claimed in the paper under consideration, the results concerning uniqueness of invariant resp. symmetrizing measures do not improve those of \textit{V. I. Bogachev} and the reviewer [J. Funct. Anal. 133, No. 1, 168-223 (1995; Zbl 0840.60069)]. In the latter paper the notion of invariant measure is much wider and does not depend on a specially chosen version of the corresponding process. Example 6.1 ibidem describes cases of non-uniqueness on \(\mathbb{R}^1\), which would contradict the uniqueness result in the paper under consideration if the two notions of invariant measures coincided.