2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221241Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $Ω$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(Ω)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.8 pages--Replacement, with corrections, of an earlier versionAnalysis of PDEs35J70, 35Hxx, 35F05, 31C15text