Flows and invariance for degenerate elliptic operators

Let S be a sub-Markovian semigroup on L ₂(ℝ ^d) generated by a self-adjoint, second-order, divergence-form, elliptic operator H with W ^1,∞(ℝ ^d) coefficients c _kl, and let Ω be an open subset of ℝ ^d. We prove that if either C ^∞ _c(ℝ ^d) is a core of the semigroup generator of the consistent semigroup on L _p(ℝ ^d) for some p ∈ [1,∞] or Ω has a locally Lipschitz boundary, then S leaves L ₂ (Ω) invariant if and only if it is invariant under the flows generated by the vector fields Σ ^d _l=1c _kl∂ _l for all k. Further, for all p ∈ [1,2] we derive sufficient conditions on the coefficients for the core property to be satisfied. Then by combination of these results we obtain various examples of invariance in terms of boundary degeneracy both for Lipschitz domains and domains with fractal boundaries.