  We study stochastic dynamics of an inverted pendulum subject to a random
force in the horizontal direction (Whitney's problem). Considered on the entire
time axis, the problem admits a unique solution that always remains in the
upper half plane. We formulate the problem of statistical description of this
never-falling trajectory and solve it by a field-theoretical technique assuming
a white-noise driving. In our approach based on the supersymmetric formalism of
Parisi and Sourlas, statistic properties of the never-falling trajectory are
expressed in terms of the zero mode of the corresponding transfer-matrix
Hamiltonian. The emerging mathematical structure is similar to that of the
Fokker-Planck equation, which however is written for the "square root" of the
probability distribution function. Our results for the statistics of the
non-falling trajectory are in perfect agreement with direct numerical
simulations of the stochastic pendulum equation. In the limit of strong driving
(no gravitation), we obtain an exact analytical solution for the instantaneous
joint probability distribution function of the pendulum's angle and its
velocity.
