  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.
