  One of the basic tenets in information theory, the data processing inequality
states that output divergence does not exceed the input divergence for any
channel. For channels without input constraints, various estimates on the
amount of such contraction are known, Dobrushin's coefficient for the total
variation being perhaps the most well-known. This work investigates channels
with average input cost constraint. It is found that while the contraction
coefficient typically equals one (no contraction), the information nevertheless
dissipates. A certain non-linear function, the \emph{Dobrushin curve} of the
channel, is proposed to quantify the amount of dissipation. Tools for
evaluating the Dobrushin curve of additive-noise channels are developed based
on coupling arguments. Some basic applications in stochastic control,
uniqueness of Gibbs measures and fundamental limits of noisy circuits are
discussed.
  As an application, it shown that in the chain of $n$ power-constrained relays
and Gaussian channels the end-to-end mutual information and maximal squared
correlation decay as $\Theta(\frac{\log\log n}{\log n})$, which is in stark
contrast with the exponential decay in chains of discrete channels. Similarly,
the behavior of noisy circuits (composed of gates with bounded fan-in) and
broadcasting of information on trees (of bounded degree) does not experience
threshold behavior in the signal-to-noise ratio (SNR). Namely, unlike the case
of discrete channels, the probability of bit error stays bounded away from
$1\over 2$ regardless of the SNR.
