# Probabilistic characterizations of essential self-adjointness and removability of singularities

## TL;DR

This paper explores the conditions under which the Laplacian and its fractional powers are essentially self-adjoint on the complement of a compact set, linking potential theory, capacities, and stochastic processes.

## Contribution

It provides a probabilistic characterization of essential self-adjointness for Laplacians with singularities, connecting classical analysis with stochastic process theory.

## Key findings

- Critical size of singular set characterized by capacities and Hausdorff measures
- Essential self-adjointness linked to Kakutani type theorems for stochastic processes
- Survey of known results on fractional Laplacians and singularities

## Abstract

We consider the Laplacian and its fractional powers of order less than one on the complement $\mathbb{R}^d\setminus\Sigma$ of a given compact set $\Sigma\subset \mathbb{R}^d$ of zero Lebesgue measure. Depending on the size of $\Sigma$, the operator under consideration, equipped with the smooth compactly supported functions on $\mathbb{R}^d \setminus \Sigma$, may or may not be essentially self-ajoint. We survey well known descriptions for the critical size of $\Sigma$ in terms of capacities and Hausdorff measures. In addition, we collect some known results for certain two-parameter stochastic processes. What we finally want to point out is, that, although a priori essential self-adjointness is not a notion directly related to classical probability, it admits a characterization via Kakutani type theorems for such processes.