On the Asymptotic Estimates for Exit Probabilities and Minimum Exit Rates of Diffusion Processes Pertaining to a Chain of Distributed Control Systems

Abstract

In this paper, we consider a diffusion process pertaining to a distributed control system formed by a chain of $n$ subsystems (with $n \ge 2$), where a random perturbation enters only in the first subsystem and is then subsequently transmitted to the other subsystems. We also assume that, for any $\ell \in \{2, \ldots, n\}$, the distributed control system formed by the first $\ell$ subsystems (i.e., from the first subsystem up to the $\ell$th subsystem) satisfies an appropriate Hörmander condition. As a result, the diffusion process is degenerate, in the sense that the backward operator associated with it is a degenerate parabolic equation. In particular, we consider the following two problems: (i) We provide an asymptotic estimate for the exit probability with which the diffusion process (corresponding to a particular subsystem) exits from a given bounded open domain during a certain time interval; the approach for such an asymptotic estimate basically relies on the interpretation of the exit probability function as a value function for a family of stochastic control problems that are associated with the underlying chain of distributed control system with small random perturbations. (ii) We establish a connection between the minimum exit rate with which the diffusion process exits from the given bounded open domain and the principal eigenvalue for the infinitesimal generator with zero boundary conditions. Such a connection also 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. Moreover, we provide an estimate for the attainable exit probability of the diffusion process with respect to a set of admissible optimal Markov controls for the optimal control problems.