Let $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued Markov process, $s, a, u >0$, $I(a) := \{[k \cdot a, (k+1) \cdot a] \ : \ k \in \mathbb{Z} \} $ the family of $a$-integral tiles covering $\mathbb{R}$. Let $M_u(a,s)$ the random variable counting the number of tiles $[k \cdot a, (k+1) \cdot a]$ in $I(a)$ hit by $(X_t)$ at some time $t \in [u,u+s]$. The notion is introduced in the paper "Hausdorff Dimension Theorems for Self-Similar Markov Processes" by Luqin Liu and Yimin Xiao in lemma 3.1 for Markov processes.
My question is if for $(X_t)_{t \in [0, \infty)}$ a $\mathbb{R}$- valued Lévy process the expectation value of $M_u(a,s)$ is "translation invariant" in the sense that
$$\mathbb{E}[M_u(a,s)]= \mathbb{E}[M_0(a,s)] $$
i.e. the expectation value of the number of hit tiles $[k \cdot a, (k+1) \cdot a]$ in $I(a)$ from $I(a)$ by $(X_t)$ at some time $t \in [u,u+s]$ is the same as the expectation value of the number of hit by $(X_t)$ at some time $t \in [0,s]$?