We consider the Laplacian and its fractional powers of order less than one on the complement $\mathbb{R}^d\setminusΣ$ of a given compact set $Σ\subset \mathbb{R}^d$ of zero Lebesgue measure. Depending on the size of $Σ$, the operator under consideration, equipped with the smooth compactly supported functions on $\mathbb{R}^d \setminus Σ$, may or may not be essentially self-ajoint. We survey well known descriptions for the critical size of $Σ$ 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.