  Statistical description of stochastic dynamics in highly unstable potentials
is strongly affected by properties of divergent trajectories, that quickly
leave meta-stable regions of the potential landscape and never return. Using
ideas from theory of Q-processes and quasi-stationary distributions, we analyze
position statistics of non-diverging trajectories. We discuss two limit
distributions which can be considered as (formal) generalizations of the Gibbs
canonical distribution to highly unstable systems. Even though the associated
effective potentials differ only slightly, properties of the two distributions
are fundamentally different for all highly unstable system. The distribution
for trajectories conditioned to diverge in an infinitely distant future is
localized and light-tailed. The other distribution, describing trajectories
surviving in the meta-stable region at the instant of conditioning, is
heavy-tailed. The exponent of the corresponding power-law tail is determined by
the leading divergent term of the unstable potential. We discuss different
equivalent forms of the two distributions and derive properties of the
effective statistical force arising in the ensemble of non-diverging
trajectories after the Doob h-transform. The obtained explicit results
generically apply to non-linear dynamical models with meta-stable states and
fast kinetic transitions.
