On the Attainable Distributions of Diffusion Processes Pertaining to a Chain of Distributed Systems

In this talk, we consider a diffusion process pertaining to a chain of distributed systems with random perturbations that satisfies a weak Hörmander type condition. In particular, we consider the following stochastic control problem with two objectives. The first one being of a reachability-type that consists of determining the set of attainable distribution laws at a given time starting from an initial distribution law; while the second one involves minimizing the relative entropy of the attainable distribution law with respect to the initial law. Using the logarithmic transformations approach introduced by Fleming (e.g., see [5] and [6]), we provide a sufficient condition on the existence of an optimal admissible control for such a stochastic control problem which is amounted to changing the drift term by a certain perturbation suggested by Jamison in the context of reciprocal processes (e.g., see [8] and [9] (cf. [3])). Moreover, such a perturbation coincides with a minimum energy control among all admissible controls forcing the diffusion process to the desired attainable distribution law starting from the initial law. Finally, using measure transform techniques (e.g., see [7], [4] or [10]), we characterize the most probable path-space for the diffusion process corresponding to such changes in the drift term of the distributed systems.