Hitting times for the perturbed reflecting random walk

We consider a nearest neighbor random walk on Z which is reflecting at 0 and perturbed when it reaches its maximum. We compute the law of the hitting times and derive many corollaries, especially invariance principles with (rather) explicit descriptions of the asymptotic laws. We also obtain some results on the almost sure asymptotic behavior. As a by-product one can derive results on the reflecting Brownian motion perturbed at its maximum.