We propose a new field of study—basing ourselves on Hairer–Mirzakhani—uniting stochastic partial differential equations (SPDEs) with hyperbolic flows on moduli spaces of Riemann surfaces. Our construction embeds singular SPDE dynamics, such as the Kardar–Parisi–Zhang (KPZ) equation, into the Teichmüller and Weil–Petersson dynamical frameworks studied by Mirzakhani. This hybrid system models photon dynamics by combining probabilistic irregularities with deterministic hyperbolic geometry. We establish existence and uniqueness of solutions to stochastic flows on moduli space Mg, prove the existence of invariant measures absolutely continuous with respect to Weil–Petersson volume, and define a stochastic entropy functional that governs photon decoherence. As an example, we develop a stochastic KPZ flow on Teichmüller space and prove mixing and entropy production properties. Our results initiate a new field of study: stochastic–geometric quantum dynamics.