in which cases this definition coincides with the Definition \[DefWalkD\].

\[SkipfreeP\] For all $t \geq 0$, we have $\mathbb{P}_x$-almost everywhere, that $X_t \in {\operatorname{supp}}(m)$.

We have $\left((X_t )_{t \geq 0}, (\mathcal{F}_{\hat{\Phi}^{-1}_t})_{t \geq 0}, (\theta_{\hat{\Phi}^{-1}_t} )_{t \geq 0}, ( \mathbb{P}_x )_{x \in {\operatorname{supp}}(m)}\right)$ defines a Feller process. Namely, the tuple defines is a strong Markov process such that, for all $f \in C_b ({\operatorname{supp}}(m) )$, the function $x \mapsto \mathbb{E}_x[f(X_t) ]$ belongs to $C_b({\operatorname{supp}}(m) )$ and $\lim_{ t \downarrow 0} \mathbb{E}_x[f(X_t)] =f(x)$. Here, $C_b ({\operatorname{supp}}(m) )$ denotes the set of bounded continuous function with domain ${\operatorname{supp}}(m)$.

To compute the walk dimension and infinitesimal generator, we need the following lemma.

\[PotentialGapD\] Fix $x_0,x_1 \in {\operatorname{supp}}(m)$ with $x_0<x_1$ and set $\tau_{(x_0,x_1)} \coloneqq \inf\{ t \geq 0 \colon X_t \notin (x_0,x_1)\}$. For $x \in (x_0,x_1) \cap {\operatorname{supp}}(m)$, the following holds.

1.  For all $t \geq 0$ and all bounded measurable functions $f$ on ${\operatorname{supp}}(m)$ we have $$\begin{aligned}
    \qquad\quad\mathbb{E}_x\left[ \int_0^{\tau_{[x_0,x_1]}}f(X_s)\mathrm{d}s \right]=\int_{[x_0,x_1]} G_{x_0,x_1}(x,y)f(y)\;{\mathrm{d}m}(y),
    \quad \text{where} \quad
    G_{x_0,x_1}(x,y) \coloneqq  2\dfrac{(x \wedge y-x_0)(x_1-x \vee y)}{x_1-x_0}
    $$ for $x,y \in (x_0,x_1)$. In particular, we obtain $\mathbb{E}_x [ \tau_{(x_0,x_1)} ] < \infty$.

2.  We have $$\begin{aligned}
    \mathbb{P}_x\left(X_{\tau_{(x_0,x_1)}} =x_0\right)=\dfrac{x_1-x}{x_1-x_0}
    \quad \text{and} \quad
    \mathbb{P}_x\left(X_{\tau_{(x_0,x_1)}} =x_1\right)=\dfrac{x-x_0}{x_1-x_0}.\end{aligned}$$

3.  We have for $x<x_1$ with $x,x_1 