Large deviations for random walk in random environment with holding times

Suppose that the integers are assigned the random variables {ω_x, μ_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 μ_x and then, after one unit of time, moves one step to the right with probability ω_x, and one step to the left with probability 1 - ω_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.