  We formulate a holographic Wilsonian renormalization group flow for strongly
coupled systems with a gravity dual, motivated by the need to extract
efficiently low energy behavior of such systems. Starting with field theories
defined on a cut-off surface in a bulk spacetime, we propose that integrating
out high energy modes in the field theory should correspond to integrating out
a part of the bulk geometry. We describe how to carry out this procedure in
practice in the classical gravity approximation using examples of scalar and
vector fields. By integrating out bulk degrees of freedom all the way to a
black hole horizon, this formulation defines a refined version of the black
hole membrane paradigm. Furthermore, it also provides a derivation of the
semi-holographic description of low energy physics.
