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

Abstract

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. Assuming a white-noise driving, we develop a field-theoretical approach to statistical description of this never-falling trajectory based on the supersymmetric formalism of Parisi and Sourlas. 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. In the limit of strong driving, we obtain an exact analytical solution for the instantaneous joint distribution function of the pendulum's angle and its velocity.