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We will discuss the non-equilibrium dynamics of a low
temperature 2-dim black hole coupled to an external bath, using the dual
SYK model. Tracing over the bath degrees of freedom, in the
Schwinger-Keldysh formalism, leads to solving non-local stochastic
differential equations for the BH degrees of freedom. We describe the
combination of SYK operators that enable complete evaporation of the BH,
and the fluctuations that characterise the end point of the evaporation
process. This talk is based on JHEP 08 (2023) 171 (arXiv:2210.15579).