We will discuss the non-equilibrium dynamics of a 2-dim black hole coupled to an external bath, using the Schwinger-Keldysh formalism. Tracing the evaporation in real time amounts to solving non-local stochastic differential equations for the BH degrees of freedom. We describe the fluctuations that characterise the end point of the evaporation process. This talk is based on arXiv:2210.15579.