  Let $\mathcal{V}$ be a complete discrete valuation ring of unequal
characteristic with perfect residue field, $\mathcal{P}$ be a smooth,
quasi-compact, separated formal scheme over $\mathcal{V}$, $\mathcal{Z}$ be a
strict normal crossing divisor of $\mathcal{P}$ and $\mathcal{P}^\sharp :=
(\mathcal{P}, \mathcal{Z})$ the induced smooth formal log-scheme over
$\mathcal{V}$. In Berthelot's theory of arithmetic $\mathcal{D}$-modules, we
work with the inductive system of sheaves of rings
$\smash{\hat{\mathcal{D}}}_{\mathcal{P} ^\sharp} ^{(\bullet)} :=
(\smash{\hat{\mathcal{D}}}_{\mathcal{P}^\sharp} ^{(m)})_{m\in \mathbb{N}}$,
where $\smash{\hat{\mathcal{D}}}_{\mathcal{P}^{\sharp}} ^{(m)}$ is the $p$-adic
completion of the ring of differential operators of level $m$ over
$\mathcal{P}^{\sharp}$. Moreover, he introduced the sheaf $\mathcal{D}
^\dagger_{\mathcal{P}
^{\sharp},\mathbb{Q}}:=\underset{\underset{m}{\longrightarrow}}{\lim}\,
\smash{\hat{\mathcal{D}}}_{\mathcal{P}} ^{(m)} \otimes_{\mathbb{Z}}\mathbb{Q}$
of differential operators over $\mathcal{P}$ of finite level. In this paper, we
define the notion of overcoherence for complexes of
$\smash{\hat{\mathcal{D}}}_{\mathcal{P} ^{\sharp}} ^{(\bullet)} $-modules and
check that this notion is compatible to that of overcoherence for complexes of
$\mathcal{D} ^\dagger_{\mathcal{P},\mathbb{Q}}$-modules.
