the only terms generated in a Gaussian theory which the linear models under discussion are), and one obtains the $\beta$-functions for the double trace couplings along the flow. In the present context however what we have is a situation wherein we are forced to engineer a specific double trace deformation on the boundary so as to ensure that we satisfy the Dirichlet boundary conditions on $\Sigma_D$. This is conceptually different from usual notions of RG, where one does not conventionally consider state dependent boundary conditions in the UV. However, there is a sense in which renormalisation of sources takes place which will become quite clear when we look at the gravitational problem.

The Dirichlet problem for gravity {#s:dgrav}
=================================

Having understood the boundary meaning of the Dirichlet problem for probe fields in a fixed background, we now turn to the situation where we consider dynamical gravity in the bulk. While we could consider other matter degrees of freedom in the bulk whose backreaction we now have to take into account, we choose for simplicity to restrict attention to the dynamics in the pure gravity sector which, as is well known, is a consistent truncation of the supergravity equations of motion. 