the associated Liouville Brownian motions for generalised Kreĭn-Feller operators to that of the classical Kreĭn-Feller operators with a special focus on the concept of walk dimension. This complements and partially resembles the general framework established by Dynkin [@DynkinI_II Vol.I,6].

Setup and statement of main results
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Our setting
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Let $\mu$ and $\nu$ denote two Borel probability measures on $[0,1]$ with ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$, $\nu(\{0,1\})=0$ and $\mu$ atomless. Denote the distribution function of $\mu$ and $\nu$ by $F_{\mu}$ and $F_{\nu}$, respectively. Let $(C_{\nu, \mu}, \lVert \cdot \rVert_{\infty} )$ denote the Banach space of continuous function on $[0,1]$ which are linear in scale $F_{\mu}$ on intervals where $F_{\nu}$ is constant. Namely, on each connected component $J$ of $[0,1] \setminus {\operatorname{supp}}(\nu)$ the function $f$ is linear in $F_{\mu}$, that is $f(x)=a_J F_{\mu}(x)+b_J$ for all $x \in J$ and some $a_J, b_J \in  \mathbb{R}$. As indicated above, we let $\Lambda$ denote the one-dimensional Lebesgue measure restricted to $[0,1]$.

Set $\S^w \coloneqq L^2(\nu)$ and $\S^s \coloneqq C_{\nu,\mu}$, where $w$ stands for [*weak*]{} and $s$ stands for [*strong*]{}; we sometimes write $\S^*(\mu,\nu)$ instead of $\S^*$ to stress the dependence of the underlying measure spaces for $* \in \{s,w\}$. In what follows we will mainly be 