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As we have reviewed above, various results exist in literature that suggest that solving such a Dirichlet problem is analogous to some kind of RG from the CFT point of view. Despite the strongly suggestive nature of this holographic RG point of view, it is also not very clear what kind of an RG is one speaking of within a CFT. A-priori, for one, it does not seem like the RG flow that arises from cutting off a CFT a la Wilson is the correct way to dualize the Dirichlet problem. Hence, our question - what is the CFT dual of a bulk Dirichlet problem?[^8]

As a warm-up we first consider the bulk Dirichlet problem for linear PDEs, using the simple setting of a Klein-Gordon field propagating in a cut-off AdS spacetime. In this case it is not hard to see that one is deforming the field theory by a non-local double-trace operator, whose precise form, we argue, can be extracted by suitable convolution of appropriate bulk-to-boundary propagators.

We then turn to the issue of setting up the problem in a gravitational setting, outlining it in general before moving on to the tractable setting of the fluid/gravity regime. 