Markov uniqueness of degenerate elliptic operators

Let Ω be an open subset of R ^d and H _Ω - Σ ^d _{i,j 1} ∂ _iij ∂ _j be a second-order partial differential operator on L ₂(Ω) with domain C ^∞ ₀(Ω), where the coefficients c _ij ε W ^1,∞(Ω) are real symmetric and C = (c _ij) is a strictly positive-definite matrix over Ω. In particular, H _Ω is locally strongly elliptic. We analyze the submarkovian extensions of H _Ω,i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H _Ωis Markov unique, i.e., it has a unique submarkovian extension, if and only if cap _Ω(∂Ω) 0 where cap _Ω (∂Ω) is the capacity of the boundary of Ω measured with respect to H _Ω. The second main result shows that Markov uniqueness of H _Ω is equivalent to the semigroup generated by the Friedrichs extension of H _Ω being conservative.