Entropy, Reversible Diffusion Processes, and Markov Uniqueness

Consider a symmetric bilinear form Eϕdefined on C∞c(Rd) by[formula]In this paper we study the stochastic process associated with the smallest closed markovian extension of (Eϕ, C∞c), and give a new proof of Markov uniqueness (i.e. the uniqueness of a closed markovian extension) based on purely probabilistic arguments. We also give another purely analytic one. As a consequence, we show that all invariant measures are reversible, provided they are of finite energy. The problem of uniqueness of such measures is also partially solved.