Ellipticity and Ergodicity

Abstract

Let $S=\{S_t{\}}_{t\ge 0}$ 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 $\hat{I}\text{c◯}$ 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 $L_2(\hat{I}\text{c◯})$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i={\sum }_{j=1}^d{c}_{ij}{\partial }_j$.

8 pages--Replacement, with corrections, of an earlier version