Syst\`emes inductifs surcoh\'erents de D-modules arithm\'etiques logarithmiques

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.