There is an interpretation of D-modules over "sufficiently nice" prestacks $X$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I.4 here and this paper), in which one views D-modules as ind-coherent sheaves (i.e. filtered colimits of coherent sheaves) over so-called de Rham spaces $X_{dR}$ attached to the previously mentioned "nice" prestacks $X$. As I understand it, this is essentially using the fact that each crystal in quasi-coherent sheaves comes canonically equipped with a flat connection, and thus can be seen as a D-module; the approach by Gaitsgory-Rozenblyum is therefore a version of infinitesimal cohomology (in the sense of Grothendieck-Ogus) wherein establishing the six functors is somewhat easier, as now the six functors for D-modules can be deduced from the general theory of ind-coherent sheaves. There is, however, a caveat: our prestack $X$ has to be an object in characteristic $0$, and preferably over a field of characteristic $0$, as smooth schemes over fields are automatically reduced.
Now I am aware of the fact that there is also a theory of "arithmetic" D-modules, developed by Berthelot, wherein one replaces infinitesimal sites and all the businesses involving de Rham spaces with crystalline sites, whose objects are pd-immersions and whose coverage is the usual Zariski coverage. Given the somewhat ad hoc definition of pd-structures, is it also possible (at least in principle) to reformulate the theory of arithmetic D-modules in the style of Gaitsgory and Rozenblyum ? Have there been any attempts at this, and if this is not possible, why so ?
Best Answer
The following link to a set of notes seems to contain I was looking for. It's about so-called crystalline spaces associated to proper and separated smooth schemes (which are very similar to de Rham spaces of smooth schemes) and how arithmetic D-modules can be realised as quasi-coherent sheaves on these spaces, much like how D-modules in characteristic $0$ can be thought of as quasi-coherent sheaves on de Rham spaces (I'm brushing a lot of technicalities under the rug here).
In particular, propositions 7.3 and 7.5 seem to be the result that I was looking for.
R. Gregoric, "Crystalline spaces"