The stochastic analysis of dynamic systems moving through random fields

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 180° turn (i.e., an "over-and-back" trajectory) we develop a Markovian model that involves a change in the dimension of the state after the turn. For this case we also briefly discuss the estimation problem using recently developed results on "real-time smoothing."