Markov uniqueness of degenerate elliptic operators

Abstract

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

Markovian extensions of symmetric second order elliptic differential operators

“…By using the results contained in [23], [24] and [7], we expect that our results can be further generalized to hold on domains with a Lipschitz boundary. Lastly let us notice that the Fukushima-Watanabe theorem has been recently extended by Robinson and Sikora (see [45], Theorem 1.1) to the case of an elliptic operator on an arbitrary open bounded set Ω; the final goal should be to generalize the results here presented to the case of such sets, replacing the topological boundary with the Martin one. • Given an orthogonal projector Π : h → h, we use the same symbol Π to denote both the injection Π|R(Π) : R(Π) → h and the surjection (Π|R(Π)) * : h → R(Π); • E(h) denotes the set of couples (Π, Θ), where Π is an orthogonal projection in h and Θ is a self-adjoint operator in the Hilbert space R(Π);…”

Uniqueness of diffusion on domains with rough boundaries

“…. But this criterion is a boundary condition and in earlier papers [RS11a] [RS11b] [Rob13] it was established that it is equivalent to Γ having zero capacity relative to the form h 0 . These earlier results were stated for forms with coefficients c kl ∈ W 1,∞ (Ω) or W 1,∞ loc (Ω) but in fact no smoothness of the coefficients is necessary.…”

“…Then h N is a Dirichlet form and the associated operator H N is the extension of H 0 corresponding to generalized Neumann boundary conditions. But if k is a general Dirichlet form extension of h 0 then D(h D ) ⊆ D(k) ⊆ D(h N ) (see [FOT94], Section 3.3, [RS11a], Theorem 1.1, or [RS11b], Theorem 2.1). Thus h N ≤ k ≤ h D in the sense of ordering of quadratic forms.…”

“…Thus h 0 is Markov unique if and only if h D = h N . It was established in [RS11a] [RS11b] (see also Section 2) that this latter condition is equivalent to the boundary ∂Ω having capacity zero measured with respect to the form h N . This criterion is a property which depends on the degeneracy of the coefficients c kl near the boundary together with the regularity and uniformity properties of ∂Ω.…”

Uniqueness of form extensions and domination of semigroups

“…For divergence type forms without potential a characterization of E (N) = E (D) is given in [RS11] (also compare [GM13] for analog results on weighted manifolds) in terms of the capacity of the boundary: For a measurable set A ⊂ Ω the capacity is defined as This result has been generalized to more than one dimension and improved in various directions. We just mention as one example the conditions given in Thm.…”