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We prove that the dimension drop phenomenon holds for the harmonic measureassociated to a transient random walk in a random environment (as defined by R.Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on aninfinite Galton-Watson tree without leaves. We use regeneration times andergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peresin 1996 to give an explicit construction of the invariant measure for theforward environment seen by the particule at exit times which is absolutelycontinuous with respect to the joint law of the tree and the path of the randomwalk.

Dimension Drop for Transient Random Walks on Galton-Watson Trees in Random Environments