Regular subspaces of symmetric stable processes

TL;DR

This work characterizes when a 1-D symmetric $oldsymbol{eta}$-stable Dirichlet form admits proper regular subspaces and provides a concrete scale-function-based description for the $oldsymbol{eta} ext{ in }(1,2]$ case. The authors show a sharp dichotomy: no proper regular subspaces for $oldsymbol{eta} obreak<1$, and a full characterization for $oldsymbol{eta} obreak ext{ in }[1,2]$ via scale functions $s$ in the local Sobolev class $oldsymbol{ extbf{S}}^{rac{eta}{2}}_{ ext{loc}}$, with $igar{oldsymbol{ ext{F}}_sig floor$ realized as a regular subspace of $H^{rac{eta}{2}}$ and properness tied to capacity and quasi-continuity. The paper extends these insights to general symmetric Lévy processes, showing finite-variation processes admit no proper subspaces, while non-finite-variation cases relate to capacity concentration on the scale's support. The approach blends ladder-function decompositions, capacity theory, and scale-function transforms to connect analytic subspace structure with probabilistic path properties, offering a constructive method to generate proper subspaces when possible.

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 $α$-stable processes are considered. The main result is that it admits proper regular subspaces if and only if $α\in [1,2]$. Moreover, for $α\in(1,2)$, 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.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2