Flows and invariance for elliptic operators

Abstract

Let $S$ be the submarkovian semigroup on $L_2({\bf R}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty}$ coefficients $c_{kl}$. Further let $\Omega$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $\sum_{l=1}^d c_{kl} \partial_l$ for all $k$.