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We investigate the multi-step prediction of the drivable space, represented by Occupancy Grid Maps (OGMs), for autonomous vehicles. Our motivation is that accurate multi-step prediction of the drivable space can efficiently improve path planning and navigation resulting in safe, comfortable and optimum paths in autonomous driving. We train a variety of Recurrent Neural Network (RNN) based architectures on the OGM sequences from the KITTI dataset. The results demonstrate significant improvement of the prediction accuracy using our proposed difference learning method, incorporating motion related features, over the state of the art. We remove the egomotion from the OGM sequences by transforming them into a common frame. Although in the transformed sequences the KITTI dataset is heavily biased toward static objects, by learning the difference between consecutive OGMs, our proposed method provides accurate prediction over both the static and moving objects.