Probabilistic Characterizations of Essential Self-Adjointness and Removability of Singularities

Abstract

Abstract. We consider the Laplacian and its fractional powers of order less than one on the complement R ∖ Σ of a given compact set Σ ⊂ R of zero Lebesgue measure. Depending on the size of Σ, the operator under consideration, equipped with the smooth compactly supported functions on R ∖ 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.

Capacities, Removable Sets and Lp-Uniqueness on Wiener Spaces

“…Here we are interested in L p -uniqueness after the removal of a small closed set Σ ⊂ B of zero measure. This is similar to our discussion in [25] and, in a sense, similar to a removable singularities problem, see for instance [35] or [36] or [2,Section 2.7].…”

“…Théorème] and [37], see also [2, Theorem 5.1.13]. Combined with Theorem 5.2 this yields a necessary codimension condition which is similar as in the case of Laplacians on Euclidean spaces, [5,25]. If (−(−L) m , F C ∞ b (N)) is L p -unique, then ̺ d (Σ) = 0 for all d < 2mp.…”

“…The spaces B and F × F are isomorphic under the map p F × (I − p F ). If A ⊂ B is analytical and for any x ∈ F the section with respect to the above product is denotes by A x ⊂ F , then for any a ∈ R the set {x ∈ F : θ [20, 32. Théorème] and [37], see also [ Combined with Theorem 5.2 this yields a necessary codimension condition which is similar as in the case of Laplacians on Euclidean spaces, [5,25]. A more causal connection between uniqueness problems for operators and classical probability should involve certain branching diffusions rather than multiparameter processes, but even for finite dimensional Euclidean spaces the problem is not fully settled and remains a future project.…”

“…For the Laplacian on R n one main observation is that, if a compact set Σ of zero measure is removed from R n , the essential self-adjointness of (∆, C ∞ c (R n \ Σ)) in L 2 (R n ) implies that dim H Σ ≤ n − 4, where dim H denotes the Hausdorff dimension. See [5,Theorems 10.3 and 10.5] or [25,Theorem 2]. This necessary 'codimension four' condition can be rephrased by saying that we must have H n−d (Σ) = 0 for all d < 4, where H n−d denotes the Hausdorff measure of dimension n − d.…”

“…It is well known that this operator is essential self-adjoint in L 2 (R n ) if and only if n ≥ 4, [47, p.114] and [39,Theorem X.11,p.161]. Generalizations of this example to manifolds have been provided in [12] and [34], more general examples on Euclidean spaces can be found in [5] and [25], further generalizations to manifolds and metric measure spaces will be discussed in [26]. For the Laplacian on R n one main observation is that, if a compact set Σ of zero measure is removed from R n , the essential self-adjointness of (∆, C ∞ c (R n \ Σ)) in L 2 (R n ) implies that dim H Σ ≤ n − 4, where dim H denotes the Hausdorff dimension.…”