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In this paper we consider dynamic systems that move along specified trajectories across random fields, where the field acts as a driving force to the dynamic system. For a specific class of random fields we develop equations for the evolution of the covariance of the state of the dynamic system, and in the special case in which the trajectory is a straight line path followed by a 1800 turn (i.e. an "over and back" trajectory) we develop a Markovian model that involves a change in the dimensiondoi:10.21236/ada080204 fatcat:r43au33wgrc45bjha7vk4nrjxi