The narrow escape problem deals with the calculation of the mean escape time (MET) of a Brownian particle from a bounded domain through a small hole on the domain's boundary. Here we develop a formalism that allows us to evaluate the non-escape probability of a gas of diffusing particles that may interact with each other. In some cases the non-escape probability allows us to evaluate the MET of the first particle. The formalism is based on the fluctuating hydrodynamics and the recently

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... macroscopic fluctuation theory. We also uncover an unexpected connection between the narrow escape of interacting particles and thermal runaway in chemical reactors.