  We discuss the statistics of first-passage times of a Brownian particle
moving in a highly unstable nonlinear potential proportional to an odd power of
position. We observe temperature-induced shortening of the mean first-passage
time and its dependence on the power of nonlinearity. We propose a passage-time
fraction as both a simple and experimentally detectable witness of the
nonlinearity. It is advantageously independent of all other parameters of the
experiment and observable for a small number of trajectories. To better
characterize the stochastic passage in the unstable potential, we introduce an
analogy of the signal-to-noise ratio for the statistical distribution of the
first-passage times. Interestingly, the upper bound for the signal-to-noise
ratio is temperature independent in the unstable potential. Finally, we
describe the nonequilibrium steady state of the particle cyclically passing
through unstable odd nonlinearity. The maximum of the steady-state probability
distribution shifts against the directions of the current and this
counterintuitive effect increases with temperature. All these thermally induced
effects are very promising targets for experimental tests of highly nonlinear
stochastic dynamics of particles placed into optical potential landscapes of
shaped optical tweezers.
