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

We consider the generalised Kre\uın-Feller operator Δ_ν,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 Δ_ν,μ can be transferred to the corresponding problem for the classical Kre\uın-Feller operator Δ_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 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.