Generalised Kreĭn–Feller operators and gap diffusions via transformations of measure spaces

Marc Kesseböhmer, Aljoscha Niemann, Tony Samuel, Hendrik Weyer

Research output: Working paper/PreprintPreprint


We consider the generalised Kre\uın-Feller operator Δν,μ with respect to compactly supported Borel probability measures μ and ν under the natural restrictions supp(ν)⊆supp(μ) and μ is atomless. We show that the solutions of the eigenvalue problem for Δν,μ can be transferred to the corresponding problem for the classical Kre\uın-Feller operator Δν,Λ=∂μx with respect to the Lebesgue measure Λ via an isometric isomorphism of the underlying Banach spaces. In this way we reprove and consolidate many known results on the spectral asymptotics of Kre\uın-Feller operators. Additionally, we investigate infinitesimal generators of generalised gap diffusions associated to generalised Kre\uın-Feller operators under Neumann boundary condition and determine their scale functions and speed measures. Extending the measure μ and ν to the real line allows us to determine the walk dimension of the given gap diffusion.
Original languageEnglish
Number of pages15
Publication statusPublished - 22 Oct 2021


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