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 $$\mathcal{P}$^\sharp := ($\mathcal{P}$, $\mathcal{Z}$)$theinducedsmoothformallog-schemeover$$\mathcal{V}$$\mathrm{.}InBerthelo{t}^{\mathrm{\prime }}stheoryofarithmetic$$\mathcal{D}$$-modules, we work with the inductive system of sheaves of rings $\smash{\hat{$\mathcal{D}$}}{$\mathcal{P}$ ^\sharp} ^{(\bullet)} := (\smash{\hat{$\mathcal{D}$}}{$\mathcal{P}$^\sharp} ^{(m)}){m\in $\mathbb{N}$}$, where ${\widehat{\mathcal{D}}}_{{\mathcal{P}}^{\mathrm{♯}}}^{(m)}$ is the $p$-adic completion of the ring of differential operators of level $m$ over ${\mathcal{P}}^{\mathrm{♯}}$. Moreover, he introduced the sheaf $$\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}$$ofdifferentialoperatorsover$$\mathcal{P}$$ of finite level. In this paper, we define the notion of overcoherence for complexes of ${\widehat{\mathcal{D}}}_{{\mathcal{P}}^{\mathrm{♯}}}^{(\bullet )}$-modules and check that this notion is compatible to that of overcoherence for complexes of ${\mathcal{D}}_{\mathcal{P},\mathbb{Q}}^{†}$-modules.