  Stochastic motion of particles in a highly unstable potential generates a
number of diverging trajectories leading to undefined statistical moments of
the particle position. This makes experiments challenging and breaks down a
standard statistical analysis of unstable mechanical processes and their
applications. A newly proposed approach takes advantage of the local
characteristics of the most probable particle motion instead of the divergent
averages. We experimentally verify its theoretical predictions for a Brownian
particle moving near an inflection in a highly unstable cubic optical
potential. The most-likely position of the particle atypically shifts against
the force despite the trajectories diverge in the opposite direction. The local
uncertainty around the most-likely position saturates even for strong diffusion
and enables well-resolved position detection. Remarkably, the measured particle
distribution quickly converges to the quasi-stationary one with the same
atypical shift for different initial particle positions. The demonstrated
experimental confirmation of the theoretical predictions approves the utility
of local characteristics for highly unstable systems which can be exploited in
thermodynamic processes to uncover energetics of unstable systems.
