  We study two-receiver Poisson channels using tools derived from stochastic
calculus. We obtain a general formula for the mutual information over the
Poisson channel that allows for conditioning and the use of auxiliary random
variables. We then use this formula to compute necessary and sufficient
conditions under which one Poisson channel is less noisy and/or more capable
than another, which turn out to be distinct from the conditions under which
this ordering holds for the discretized versions of the channels. We also use
general formula to determine the capacity region of the more capable Poisson
broadcast channel with independent message sets, the more capable Poisson
wiretap channel, and the general two-decoder Poisson broadcast channel with
degraded message sets.
