  In semiclassical holographic duality, the running couplings of a field theory
are conventionally identified with the classical solutions of field equations
in the dual gravitational theory. However, this identification is unclear when
the bulk fields fluctuate. Recent work has used a Wilsonian framework to
propose an alternative identification of the running couplings in terms of
non-fluctuating data; in the classical limit, these new couplings do not
satisfy the bulk equations of motion. We study renormalization scheme
dependence in the latter formalism, and show that a scheme exists in which
couplings to single trace operators realize particular solutions to the bulk
equations of motion, in the semiclassical limit. This occurs for operators with
dimension $\Delta \notin \frac{d}{2} + \ZZ$, for sufficiently low momenta. We
then clarify the relation between the saddle point approximation to the
Wilsonian effective action ($S_W$) and boundary conditions at a cutoff surface
in AdS space. In particular, we interpret non-local multi-trace operators in
$S_W$ as arising in Lorentzian AdS space from the temporary passage of
excitations through the UV region that has been integrated out. Coarse-graining
these operators makes the action effectively local.
