  A continuous-time white Gaussian channel can be formulated using a white
Gaussian noise, and a conventional way for examining such a channel is the
sampling approach based on the Shannon-Nyquist sampling theorem, where the
original continuous-time channel is converted to an equivalent discrete-time
channel, to which a great variety of established tools and methodology can be
applied. However, one of the key issues of this scheme is that continuous-time
feedback and memory cannot be incorporated into the channel model. It turns out
that this issue can be circumvented by considering the Brownian motion
formulation of a continuous-time white Gaussian channel. Nevertheless, as
opposed to the white Gaussian noise formulation, a link that establishes the
information-theoretic connection between a continuous-time channel under the
Brownian motion formulation and its discrete-time counterparts has long been
missing. This paper is to fill this gap by establishing causality-preserving
connections between continuous-time Gaussian feedback/memory channels and their
associated discrete-time versions in the forms of sampling and approximation
theorems, which we believe will play important roles in the long run for
further developing continuous-time information theory.
  As an immediate application of the approximation theorem, we propose the
so-called approximation approach to examine continuous-time white Gaussian
channels in the point-to-point or multi-user setting. It turns out that the
approximation approach, complemented by relevant tools from stochastic
calculus, can enhance our understanding of continuous-time Gaussian channels in
terms of giving alternative and strengthened interpretation to some long-held
folklore, recovering "long known" results from new perspectives, and rigorously
establishing new results predicted by the intuition that the approximation
approach carries.
