A formula for the capacity of the general Gel´fand-Pinsker channel

We consider the Gel´fand-Pinsker problem in which the channel and state are general, i.e., possibly non-stationary, non-memoryless and non-ergodic. Using Verdú-Han´s information spectrum method and a non-trivial modification of Wyner´s piggyback coding lemma, we prove that the capacity can be expressed as an optimization over the difference of a spectral infand a spectral sup-mutual information rate. We consider various specializations including the case where the channel and state are memoryless but non-stationary. We then extend our result to obtain the capacity region of the general Gel´fand-Pinsker problem with rate-limited state information at the decoder.