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Dear Matt: The people who are actively working with this whom I know are Genady Lyubeznik and Manuel Blickle. The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$-modules (here $F$ is the Frobenius). It has been applied to show that certain local cohomology modules over regular local rings in positive characteristic have finitely many associated primes. Examples are in the following papers:

Generators of D-modules in positive characteristic

and the references there in. There is also this new preprint which might be of interest(I remember seeing some thing on your website that looks related).

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Dear Matt: The people who are actively working with this whom I know are Genady Lyubeznik and Manuel Blickle. The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$-modules (here $F$ is the Frobenius). It has been applied to show that certain local cohomology modules over regular local rings in positive characteristic have finitely many associated primes. Examples are in the following papers:

Generators of D-modules in positive characteristic

and the references there in. There is also this new preprint which might be of interest (I remember seeing some thing on your website that looks related).