Regular subspaces of symmetric stable processes

TL;DR

This paper investigates the existence and characterization of proper regular subspaces of one-dimensional symmetric alpha-stable processes, revealing that such subspaces exist if and only if alpha is between 1 and 2, and linking their existence to path variation.

Contribution

It provides a complete characterization of regular subspaces for 1D symmetric alpha-stable processes and explores the connection with path variation for general symmetric Lévy processes.

Findings

Proper regular subspaces exist if and only if alpha is in [1,2]

Characterization of regular subspaces for alpha in (1,2)

Existence of proper regular subspaces relates to finite variation of sample paths

Abstract

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 $\alpha$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $\alpha\in [1,2]$. Moreover, for $\alpha\in(1,2)$, the characterization of the regular subspaces is given. 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 its sample paths have finite variation.