  The limits of scaled relative entropies between probability distributions
associated with N-particle weakly interacting Markov processes are considered.
The convergence of such scaled relative entropies is established in various
settings. The analysis is motivated by the role relative entropy plays as a
Lyapunov function for the (linear) Kolmogorov forward equation associated with
an ergodic Markov process, and Lyapunov function properties of these scaling
limits with respect to nonlinear finite-state Markov processes are studied in
the companion paper [6].
