Conservation and invariance properties of submarkovian semigroups

TL;DR

This paper investigates the conditions under which submarkovian semigroups associated with Dirichlet forms exhibit conservation and invariance properties, linking these properties to the capacity of boundary sets and boundary conditions.

Contribution

It establishes precise equivalences between boundary capacity, boundary conditions, and conservation properties of submarkovian semigroups under general regularity assumptions.

Findings

Equality of semigroups $S_t$ and $S^D_t$ when boundary capacity is zero.

Conservation of $S$ implies conservation of $S^D$ if boundary capacity is zero.

Vanishing boundary capacity is equivalent to equality of $S^D_t$ and $S^N_t$ under certain conditions.

Abstract

Let ${\cal E}$ be a Dirichlet form on $L_2(X)$ and $\Omega$ 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 $\partial\Omega$ of $\Omega$. If $S$, $S^D$ and $S^N$ are the associated submarkovian semigroups we prove, under general assumptions of regularity and locality, that $S_t\phi = S^D_t\phi$ for all $\phi\in L_2(\Omega)$ and $t>0$ if and only if the capacity ${\mathop{\rm cap}}_\Omega(\partial\Omega)$ of $\partial \Omega$ relative to $\Omega$ is zero. Moreover, if $S$ is conservative, i.e. stochastically complete, then ${\mathop{\rm cap}}_\Omega(\partial\Omega)=0$ 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 $S^D_t \phi = S^N_t \phi$ for all $\phi\in L_2(\Omega)$ and $t>0$.