For highly unstable potentials, the long-time distribution of trajectories conditioned to never diverge is light-tailed, while the quasi-stationary distribution of trajectories surviving to a fixed time is heavy-tailed with an exponent set by the potential's divergent term.
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Living on the edge of instability
For highly unstable potentials, the long-time distribution of trajectories conditioned to never diverge is light-tailed, while the quasi-stationary distribution of trajectories surviving to a fixed time is heavy-tailed with an exponent set by the potential's divergent term.