  Molecular communication emerges as a promising communication paradigm for
nanotechnology. However, solid mathematical foundations for
information-theoretic analysis of molecular communication have not yet been
built. In particular, no one has ever proven that the channel coding theorem
applies for molecular communication, and no relationship between information
rate capacity (maximum mutual information) and code rate capacity (supremum
achievable code rate) has been established. In this paper, we focus on a major
subclass of molecular communication - the diffusion-based molecular
communication. We provide solid mathematical foundations for information theory
in diffusion-based molecular communication by creating a general
diffusion-based molecular channel model in measure-theoretic form and prove its
channel coding theorems. Various equivalence relationships between statistical
and operational definitions of channel capacity are also established, including
the most classic information rate capacity and code rate capacity. As
byproducts, we have shown that the diffusion-based molecular channel is with
"asymptotically decreasing input memory and anticipation" and "d-continuous".
Other properties of diffusion-based molecular channel such as stationarity or
ergodicity are also proven.
