Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

Abstract

We consider generalised Kreȋn-Feller operators ∆ ν,µ with respect to compactly supported Borel probability measures µ and ν under the natural restrictions supp(ν) ⊂ supp(µ) and µ atomless. We show that the solutions of the eigenvalue problem for ∆ ν,µ can be transferred to the corresponding problem for the classical Kreȋn-Feller operator ∆ ν,Λ = ∂ µ ∂ x with respect to the Lebesgue measure Λ via an isometric isomorphism of the underlying Banach spaces. In this way we reprove the spectral asymptotic on the eigenvalue counting function obtained by Freiberg. Additionally, we investigate infinitesimal generators of generalised Liouville Brownian motions associated to generalised Kreȋn-Feller operator ∆ ν,µ under Neumann boundary condition. Extending the measure µ and ν to the real line allows us to determine its walk dimension.

Spectral asymptotics of Krein--Feller operators for weak Gibbs measures on self-conformal fractals with overlaps

“…The spectral properties for the generalised case were studied in [Küc86], [Vol05] and [Fre05]. The connection between the generalised and the classical Kreȋn-Feller operator has been elaborated in [Kes+19]. In there, it has been shown that the spectral behaviour can be reduced to the classical Kreȋn-Feller operator by a straightforward transformation of measure spaces.…”