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We study stochastic dynamics of an inverted pendulum subject to a randomforce in the horizontal direction (Whitney's problem). Considered on the entiretime axis, the problem admits a unique solution that always remains in theupper half plane. We formulate the problem of statistical description of thisnever-falling trajectory and solve it by a field-theoretical technique assuminga white-noise driving. In our approach based on the supersymmetric formalism ofParisi and Sourlas, statistic properties of the never-falling trajectory areexpressed in terms of the zero mode of the corresponding transfer-matrixHamiltonian. The emerging mathematical structure is similar to that of theFokker-Planck equation, which however is written for the "square root" of theprobability distribution function. Our results for the statistics of thenon-falling trajectory are in perfect agreement with direct numericalsimulations of the stochastic pendulum equation. In the limit of strong driving(no gravitation), we obtain an exact analytical solution for the instantaneousjoint probability distribution function of the pendulum's angle and itsvelocity.

Inverted pendulum driven by a horizontal random force: statistics of the never-falling trajectory and supersymmetry