Large deviations for random walk in random environment with holding times

Suppose that the integers are assigned the random variables {w(x), mu(x)} (taking values in the unit interval times the space of probability measures on R+), which serve as an environment. This environment defines a random walk {X-t} (called a RWREH) which, when at x, waits a random time distributed according to mu(x) and then, after one unit of time, moves one step to the right with probability omega(x), and one step to the left with probability 1 - omega(x). We prove large deviation principles for X-t/ t, both quenched (i.e., conditional upon the environment), with deterministic rate function, and annealed (i.e., averaged over the environment). As an application, we show that for random walks on Galton-Watson trees, quenched and annealed rate functions along a ray differ.