Flows and invariance for elliptic operators

Abstract

Let $S$ be the submarkovian semigroup on $L_2({\mathbf{R}}^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with $W^{1,\infty }$ coefficients $c_{kl}$. Further let $\hat{I}\text{c◯}$ be an open subset of ${\mathbf{R}}^d$. Under mild conditions we prove that $S$ leaves $L_2(\hat{I}\text{c◯})$ 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 $k$.