A continuous time version of random walks in a random potential

We consider a system of continuous time random walks on Zd in a potential which is random in space and time. In spatial dimensions d > 2, and for sufficiently small random potential, we show that, as time goes to infinity, the behavior is diffusive with probability one. However, the diffusion constant is not equal to one, and is determined by the averaged process. The averaged process is found by averaging over the random potential initially. In the discrete time case the averaged process is the simple random walk; this explains why the diffusion constant is one in the discrete time case.