  We present a theoretical investigation of the stochastic dynamics of a damped
particle in a tilted periodic potential with a double well per period. By
applying the matrix continued fraction technique to the Fokker-Planck equation
in conjunction with the full counting statistics and master equation
approaches, we determine the rates of specific processes contributing to the
system's overall dynamics. At low temperatures, the system can exhibit one
running state and two distinct locked metastable states. We focus primarily on
two aspects: the dynamics of phase jumps, which are rare thermally induced
particle jumps over potential maxima, and their impact on the overall velocity
noise; and the retrapping process, involving the transition from the running to
the locked metastable states. We demonstrate the existence of fractional (in
units of $2\pi$) phase slips that differ qualitatively from conventional $2\pi$
jumps observed in single-well systems. Fractional phase slips significantly
influence the system dynamics even in regimes dominated by dichotomous-like
switching between running and locked states. Furthermore, we introduce a simple
master equation approach that proves effective in analyzing various stages of
the retrapping process. Interestingly, our analysis shows that even for a
system featuring a well-developed double-well periodic potential, there exists
a broad parameter range where the stochastic dynamics can be accurately
described by an effective single-well periodic model. The techniques introduced
here allow for valuable insights into the complex behavior of the system,
offering avenues for understanding and controlling its steady-state and
transient dynamics, which go beyond or can be complementary to direct
stochastic simulations.
