Regular subspaces of symmetric stable processes

Dongjian Qian, Jiangang Ying, Yushu Zheng

Published 2022-07-19Version 1

Roughly speaking, regular subspaces are regular Dirichlet forms which inherit the original forms with smaller domains. In this paper, regular subspaces of 1-dim symmetric $\alpha$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $\alpha\in [1,2]$. Moreover, general 1-dim symmetric L\'evy processes will also be investigated. It will be shown that whether it has proper regular subspaces is closely related to whether it has finite variation.