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We consider the exit problem for a one-dimensional system with randomswitching near an unstable equilibrium point of the averaged drift. In theinfinite switching rate limit, we show that the exit time satisfies a limittheorem with a logarithmic deterministic term and a random correctionconverging in distribution. Thus this setting is in the universality class ofthe unstable equilibrium exit under small white-noise perturbations.

Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium