Regular Subspaces of Symmetric Stable Processes

Roughly speaking, regular subspaces are regular Dirichlet forms that inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric \(\varvec{\alpha }\) $\mathit{\alpha }$ -stable processes are considered. The main result is that it admits proper regular subspaces if and only if \(\varvec{\alpha \in [1,2]}\) $\mathit{\alpha }\mathbf{\in }\mathbf{[}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{]}$ . Moreover, for \(\varvec{\alpha \in (1,2)}\) $\mathit{\alpha }\mathbf{\in }\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{)}$ , the characterization of the regular subspaces is given. General 1-dim symmetric Lévy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether its sample paths have finite variation.