  The fractional Josephson effect remains one of the decisive hallmarks of
topologically protected Majorana zero modes. We analyze the effects of parity
violating quasiparticle poisoning onto the current voltage characteristics of
topological Josephson junctions. We include poisoning events directly within
the resistively shunted junction (RSJ) model in the overdamped limit both in
the short- and long-junction regime. We calculate the current voltage
characteristics numerically where poisoning is modeled either via additional
rates in the Fokker-Planck equations or by a time dependent parity and compare
them to the limits of no and strong poisoning rates which we obtain
analytically. Combining the tilted washboard potential with poisoning events,
we show that the critical current of the long junction limit can be used as a
probe of the junction topology even in the high temperature poisoning case
where relaxation- and excitation processes are equally likely. Using the tilted
washboard potential model we develop three different schemes to measure the
poisoning rate thereby also extending the consideration to two pairs of helical
edge states containing a constriction that allows for tunneling between the two
pairs of edge states.
