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In this paper, we study McKean-Vlasov SDE living in $\mathbb{R}^d$ in thereversible case without assuming any type of convexity assumptions forconfinement or interaction potentials. Kramers' type law for the exit-time froma domain of attraction is established. Namely, in the small-noise regime, thelimit in probability of the first exit-time behaves exponentially. This resultis established using the large deviations principle as well as improvedcoupling method.  Having removed the convexity assumption, this work is a major improvement ofthe previously known results for the exit-time problem, the review of which isprovided in the paper.

Measure-dependent non-linear diffusions with superlinear drifts: asymptotic behaviour of the first exit-times