relation $ u=\delta_{K}/(\delta_{K}+\delta_{L})$.

Liouville Brownian motion and walk dimension {#sec:LBM}
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In this section we will construct Liouville Brownian motion via a time change of a fixed Brownian motion. We will give a short overview about basic properties of this stochastic process. After which, we will compute the walk dimension for special classes of gap diffusions. Additionally, we show that the infinitesimal generator coincides with a generalised Kreĭn-Feller operator.

For the tuple $\left(\Omega, \mathcal{F}, ( \mathcal{F}_t)_{t \geq 0}, ( \theta_t)_{t \geq 0}, \mathbb{P}_x\right)$ where $\mathbb{P}_x$ denotes the probability measure such that, $( \mathcal{F}_t)_{t \geq 0}$ is the right-continuous completed natural filtration and $(\theta_t )_{t \geq 0}$ is the shift-operator. The expectation with respect to $\mathbb{P}_x$ is denoted by $\mathbb{E}_x$. We call the stochastic process $(B_t )_{t \geq 0}$ a Brownian motion if the following are satisfied.

1.  $\mathbb{P}_x(B_0=x)=1$

2.  For $0 \leq s_0< \dots <s_n$ with $n \in \N$ the increments $B_{s_1}-B_{s_0}, \dots ,B_{s_n}-B_{s_{n-1}}$ are stochastically independent.

3.  The distribution of the increment $B_{s}-B_{t}$ follows a Gaussian distribution with variance $s-t$ where $s>t \geq 0$ and mean $0$.

Let $(L_{x}^t)_{t \geq 0, x \in \mathbb{R}}$ the jointly continuous version of the local time of the Brownian motion, see [@Revuz2013 Chapter 