  Planar black holes in AdS have long-lived quasinormal modes which capture the
physics of charge and momentum diffusion in the dual field theory. How should
we characterize the effective dynamics of a probe system coupled to the
conserved currents of the dual field theory? Specifically, how would such a
probe record the long-lived memory of the black hole and its Hawking
fluctuations? We address this question by exhibiting a universal gauge
invariant framework which captures the physics of stochastic diffusion in
holography: a designer scalar with a gravitational coupling governed by a
single parameter, the Markovianity index. We argue that the physics of gauge
and gravitational perturbations of a planar Schwarzschild-AdS black hole can be
efficiently captured by such designer scalars. We demonstrate that this
framework allows one to decouple, at the quadratic order, the long-lived
quasinormal and Hawking modes from the short-lived ones. It furthermore
provides a template for analyzing fluctuating open quantum field theories with
memory. In particular, we use this set-up to analyze the diffusive Hawking
photons and gravitons about a planar Schwarzschild-AdS black hole and derive
the quadratic effective action that governs fluctuating hydrodynamics of the
dual CFT. Along the way we also derive results relevant for probes of
hyperscaling violating backgrounds at finite temperature.
