Conservation and invariance properties of submarkovian semigroups

Abstract

Let $\mathcal{E}$ be a Dirichlet form on $L_2(X)$ and $\Omega$ an open subset of $X$. Then one can define Dirichlet forms ${\mathcal{E}}_D$, or ${\mathcal{E}}_N$, corresponding to $\mathcal{E}$ but with Dirichlet, or Neumann, boundary conditions imposed on the boundary $\partial \Omega$ of $\Omega$. If $S$, $S^D$ and $S^N$ are the associated submarkovian semigroups we prove, under general assumptions of regularity and locality, that $S_t\varphi =S_t^D\varphi$ for all $\varphi \in L_2(\Omega )$ and $t>0$ if and only if the capacity ${\mathrm{cap}}_{\Omega }(\partial \Omega )$ of $\partial \Omega$ relative to $\Omega$ is zero. Moreover, if $S$ is conservative, i.e. stochastically complete, then ${\mathrm{cap}}_{\Omega }(\partial \Omega )=0$ if and only if $S^D$ is conservative on $L_2(\Omega )$. Under slightly more stringent assumptions we also prove that the vanishing of the relative capacity is equivalent to $S_t^D\varphi =S_t^N\varphi$ for all $\varphi \in L_2(\Omega )$ and $t>0$.