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<div class="moz-cite-prefix">04.06.2020 19:30, alexpro пишет:<br>
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<blockquote type="cite"
cite="mid:e72a930d083235f8fbe24ced6174728d@itp.ac.ru"><br>
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<u>Nikolay Stepanov</u>, Mikhail Skvortsov
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Inverted pendulum driven by a horizontal random force: statistics
of the non-falling trajectory and supersymmetry
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We study stochastic dynamics of an inverted pendulum subject to a
random force in the horizontal direction. Considered at the entire
time axis, the problem admits a unique solution which always
remains in the upper half plane. We develop a new technique for
treating statistical properties of this unique non-falling
trajectory. In our approach based on the supersymmetric formalism
of Parisi and Sourlas, statistics of the non-falling trajectory is
expressed in terms of the zero mode of a corresponding
transfer-matrix Hamiltonian. The emerging mathematical structure
is similar to that of the Fokker-Planck equation, but it is rather
written for the «square root» of the distribution function.
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We derive the specific boundary conditions that correspond to the
non-falling trajectory. Our results for the distribution function
of the angle and its velocity at the non-falling trajectory are in
perfect agreement with direct numerical simulations of the
stochastic pendulum equation. In the limit of very strong noise,
an exact analytical solution is obtained.
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