provided in [@Fr05] for a certain class of self-similar measures. This is achieved via an application of and the identification of $\eta =  \nu \circ F_{\mu}^{-1}$ as a certain self-similar measure, in tandem with the corresponding result for the classical Kreĭn-Feller operator $\Delta_{\eta,\Lambda}$, see [@Fu87]. This generalises several of the results of [@KSW16], where the case $\nu=\mu$ was considered.

2.  Letting $(X_{t})_{t \geq 0}$ denote the a Liouville Brownian motion with speed measure $\nu \circ F_{\mu}^{-1}$ (see ), utilising our correspondence theorem (), we show that the infinitesimal generator of $(\check{F}^{-1}_{\mu}( X_t ))_{ t\geq 0}$ is the generalised Kreĭn-Feller operator $\Delta_{\nu,\mu}$ with Neumann boundary condition. Additionally, we compute the walk dimension of $(X_t)_{t \geq 0}$ and $(\check{F}^{-1}_{\mu}( X_t ))_{ t\geq 0}$.

Kreĭn-Feller operators
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Properties of classical Kreĭn-Feller operators
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In this section we collect important properties of classical Kreĭn-Feller operators, that is we consider the case $\mu= \Lambda$, with respect to weak and strong solutions. Most of these results are nowadays folklore and can be found for instance in [@MR0314125 Behauptung 2.2, Satz 2.1, Satz 3.1] with more or less detailed proofs. Since we could not find references where all the facts are proven in detail we decided to 