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

Abstract

We consider the generalised Kre\u{\i}n-Feller operator $\Delta_{\nu, \mu}$ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ under the natural restrictions $\mathrm{supp}(\nu)\subseteq\mathrm{supp}(\mu)$ and $\mu$ is atomless. We show that the solutions of the eigenvalue problem for $\Delta_{\nu, \mu}$ can be transferred to the corresponding problem for the classical Kre\u{\i}n-Feller operator $\Delta_{\nu, \Lambda}=\partial_{\mu}\partial_{x}$ with respect to the Lebesgue measure $\Lambda$ 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\u{\i}n-Feller operators. Additionally, we investigate infinitesimal generators of generalised gap diffusions associated to generalised Kre\u{\i}n-Feller operators under Neumann boundary condition and determine their scale functions and speed measures. Extending the measure $\mu$ and $\nu$ to the real line allows us to determine the walk dimension of the given gap diffusion.Comment: 15 pages, 4 figure