Geodesic
Probabilistic characterizations of essential self-adjointness and removability of singularities
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 148-162

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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. 
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     author = {M. Hinz and S.-J. Kang and J. Masamune},
     title = {Probabilistic characterizations of essential self-adjointness and removability of singularities},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
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     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a11/}
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M. Hinz; S.-J. Kang; J. Masamune. Probabilistic characterizations of essential self-adjointness and removability of singularities. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 20 (2017) no. 3, pp. 148-162. http://geodesic.mathdoc.fr/item/VVGUM_2017_20_3_a11/