Conservation and invariance properties of submarkovian semigroups

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

Conservation and invariance properties of submarkovian semigroups

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Description

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