The Stochastic-Calculus Approach to Multi-Receiver Poisson Channels

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.