Flows and invariance for elliptic operators

ter Elst, A. F. M.; Robinson, Derek W.; Sikora, Adam

Flows and invariance for elliptic operators

Authors

ter Elst, A. F. M.

Robinson, Derek W.

Sikora, Adam

Description

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 $Ω$ be an open subset of ${\bf R}^d$. Under mild conditions we prove that $S$ leaves $L_2(Ω)$ 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 $k$.