Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the dd-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location yy on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach yy grows only linearly in the distance from yy to the origin. In dimension 1 we show the existence of the asymptotic positive speed.