  Let $S=\{S_t\}_{t\geq0}$ 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 $\Omega$ 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 $L_2(\Omega)$ invariant if, and only if, it is
invariant under the flows generated by the vector fields
$Y_i=\sum^d_{j=1}c_{ij}\partial_j$.
