Conservation and invariance properties of submarkovian semigroups
| dc.creator | ter Elst, A. F. M. | |
| dc.creator | Robinson, Derek W. | |
| dc.date | 2009-03-31 | |
| dc.date.accessioned | 2026-07-07T12:58:36Z | |
| dc.date.available | 2026-07-07T12:58:36Z | |
| dc.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 $t>0$. | |
| dc.identifier | https://arxiv.org/abs/0903.5479 | |
| dc.identifier | http://arxiv.org/abs/0903.5479 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/225295 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35Hxx; 35J70; 31C15; 31C25 | |
| dc.title | Conservation and invariance properties of submarkovian semigroups | |
| dc.type | text |