This article is cited in 2 papers

Mathematics

Probabilistic characterizations of essential self-adjointness and removability of singularities

M. Hinz^a, S.-J. Kang^b, J. Masamune<sup>c

a</sup> University of Bielefeld
^b Seoul National University
^c Hokkaido University

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.

Keywords: Laplacian, essential self-adjointness, removability of singularities, probabilistic characterizations, stochastic processes.

UDC: 517
BBK: 22.161

Language: English

DOI: 10.15688/mpcm.jvolsu.2017.3.11