Markov uniqueness of degenerate elliptic operators

Abstract

Let $\hat{I}\text{c◯}$ be an open subset of $\Ri^d$ and $H_{\hat{I}}\text{c◯}=-{\sum }_{i,j=1}^d{\partial }_i{c}_{ij}{\partial }_j$ a second-order partial differential operator on $L_2(\hat{I}\text{c◯})$ with domain $C_c^{\infty }(\hat{I}\text{c◯})$ where the coefficients $c_{ij}\in W^{1,\infty }(\hat{I}\text{c◯})$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\hat{I}\text{c◯}$. In particular, $H_{\hat{I}}\text{c◯}$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_{\hat{I}}\text{c◯}$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_{\hat{I}}\text{c◯}$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_Ω(\partialΩ)=0$ where $\capp_Ω(\partialΩ)$ is the capacity of the boundary of $\hat{I}\text{c◯}$ measured with respect to $H_{\hat{I}}\text{c◯}$. The second main result establishes that Markov uniqueness of $H_{\hat{I}}\text{c◯}$ is equivalent to the semigroup generated by the Friedrichs extension of $H_{\hat{I}}\text{c◯}$ being conservative.

24 pages