  We consider random walk with bounded jumps on a hypercubic lattice of
arbitrary dimension in a dynamic random environment. The environment is
temporally independent and spatially translation invariant. We study the rate
functions of the level-3 averaged and quenched large deviation principles from
the point of view of the particle. In the averaged case the rate function is a
specific relative entropy, while in the quenched case it is a Donsker-Varadhan
type relative entropy for Markov processes. We relate these entropies to each
other and seek to identify the minimizers of the level-3 to level-1
contractions in both settings. Motivation for this work comes from variational
descriptions of the quenched free energy of directed polymer models where the
same Markov process entropy appears.
