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Let S be a sub-Markovian semigroup on L 2 (R d ) generated by a self-adjoint, second-order, divergenceform, 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 d l=1 c kldoi:10.1017/s1446788711001315 fatcat:ufydm74uobgehimodekyy5e4de