Generalised Krein-Feller operators and Liouville Brownian motion via transformations of measure spaces

Marc Kesseböhmer, Aljoscha Niemann, Tony Samuel, Hendrik Weyer

Published 2019-09-19Version 1

We consider generalised Kre\u{\i}n-Feller operators $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ under the natural restrictions $\mathrm{supp}(\nu)\subset\mathrm{supp}(\mu)$ and $\mu$ 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 the spectral asymptotic on the eigenvalue counting function obtained by Freiberg. Additionally, we investigate infinitesimal generators of generalised Liouville Brownian motions associated to generalised Kre\u{\i}n-Feller operator $\Delta_{\nu, \mu}$ under von Neumann boundary condition. Extending the measure $\mu$ and $\nu$ to the real line allows us to determine its walk dimension.