On the Minimum Exit Rate for a Diffusion Process Pertaining to a Chain of Distributed Control Systems with Random Perturbations

In this paper, we consider the problem of minimizing the exit rate with which a diffusion process pertaining to a chain of distributed control systems, with random perturbations, exits from a given bounded open domain. In particular, we consider a chain of distributed control systems that are formed by n subsystems (with n ≥ 2), where the random perturbation enters only in the first subsystem and is then subsequently transmitted to the other subsystems. Furthermore, we assume that, for any ` ∈ {2, . . . , n}, the distributed control systems, which is formed by the first ` subsystems, satisfies an appropriate Hörmander condition. As a result of this, the diffusion process is degenerate, in the sense that the infinitesimal generator associated with it is a degenerate parabolic equation. Our interest is to establish a connection between the minimum exit rate with which the diffusion process exits from the given domain and the principal eigenvalue for the infinitesimal generator with zero boundary conditions. Such a connection allows us to derive a family of Hamilton-Jacobi-Bellman equations for which we provide a verification theorem that shows the validity of the corresponding optimal control problems. Finally, we provide an estimate on the attainable exit probability of the diffusion process with respect to a set of admissible (optimal) Markov controls for the optimal control problems.