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abstract: 'We consider generalised Kreĭn-Feller operators $\Delta_{\nu, \mu}  $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ under the natural restrictions ${\operatorname{supp}}(\nu)\subset{\operatorname{supp}}(\mu)$ 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 Kreĭn-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 Kreĭn-Feller operator $\Delta_{\nu, \mu}$ under Neumann boundary condition. Extending the measure $\mu$ and $\nu$ to the real line allows us to determine its walk dimension.'
address:
- 'FB 3 – Mathematik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany'
- 'School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK'
author:
- Marc Kesseböhmer
- Aljoscha Niemann
- Tony Samuel
- Hendrik Weyer
title: 'Generalised Kreĭn-Feller operators and Liouville Brownian motion via transformations of measure spaces'
---

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Introduction and motivation
===========================

The classical Kreĭn-Feller differential operator $\Delta_{\nu, \Lambda}$, introduced in [@Fe57; @KK68], where $\nu$ denotes a non-atomic compactly 