VI].

Let $m$ denote a non-zero Borel measure on $(\R,\mathcal{B})$.

\[LBM\] We define for $ t \geq 0$ the [*(inverse) time-change function*]{} $$\begin{aligned}
\Phi_t \coloneqq \int_{\R} L_x^t \;{\mathrm{d}m}(x),\;\; \ \hat{\Phi}^{-1}_t \coloneqq \inf\left\{s \geq 0 \colon \Phi_s>t\right \}\end{aligned}$$ where $\hat{\Phi}^{-1}_t$ is called right-continuous inverse of $\Phi_t$. Now, we define the new process for $x \in {\operatorname{supp}}(m)$ $$\begin{aligned}
\left((X_t)_{ t \geq 0} \coloneqq (B_{\hat{\Phi}^{-1}_t})_{t \geq 0}, (\mathcal{F}_{\hat{\Phi}^{-1}_t})_{t \geq 0},  \mathbb{P}_x\right )\end{aligned}$$ which will be called $m$-Liouville Brownian motion with speed measure $m$ starting in $x$.

\[DefWalkD\] Let $(X_t)_{t \geq 0}$ a Liouville Brownian motion with speed measure $m$. Then the *walk dimension* $d_w(x)$ in $x \in {\operatorname{supp}}(m)$ is defined by $$\begin{aligned}
\label{WalkD}
d_W(x) \coloneqq 
\lim_{R \to \infty} \dfrac{\log(\mathbb{E}_x[\tau_{(-R+x,R+x)}]  )}{\log(R)}\end{aligned}$$ assuming that the limit exists, where $\tau_{(-R+x,R+x)} \coloneqq  \inf \{ t \geq 0 \colon X_t \notin (-R+x,R+x)\}$

Assuming the support of $m$ is bounded, then the walk dimension does not exist. In this case it can be useful to consider the so-called *local walk dimension* $$\begin{aligned}
d_{LW}(x) \coloneqq    \lim_{R \downarrow 0} \dfrac{\log(\mathbb{E}_x[\tau_{(-R+x,R+x)}]  )}{\log(R)}.\end{aligned}$$ There are other definitions of the walk dimension; for example in [@KGOLMANKHANEH2018960] the walk dimension $d_{w}(x)$ is defined by $$\begin{aligned}
\mathbb{E}_x [ (X_t-x  )^2  ] \asymp t^{2/d_w(x)}.\end{aligned}$$

It remains open 