  We consider the continuity equation for open chaotic quantum systems in the
semiclassical limit. First we explicitly calculate a semiclassical expansion
for the probability current density using an expression based on classical
trajectories. The current density is related to the survival probability via
the continuity equation, and we show that this relation is satisfied within the
semiclassical approximation to all orders. For this we develop recursion
relation arguments which connect the trajectory structures involved for the
survival probability, which travel from one point in the bulk to another, to
those structures involved for the current density, which travel from the bulk
to the lead. The current density can also be linked, via another continuity
equation, to a correlation function of the scattering matrix whose
semiclassical approximation is expressed in terms of trajectories that start
and end in the lead. We also show that this continuity equation holds to all
orders.
