Effective field theory of stochastic diffusion from gravity

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