Robustness and Approximation of Escape Times and Large Deviations Estimates for Systems with Small Noise Effects

For the purposes of estimating escape time from a given set, or other statistical properties of systems with small noise effects, it is generally assumed in applications that the system noise is white Gaussian.The Gaussian assumption greatly simplifies the computation, but is not adequate for many important classes of applications in control and communication theory. For example, when the noise is small, the mean escape time from a set can be quite sensitive to the underlying statistics even though, in the study of the effects of the noise over any fixed finite time interval, the Gaussian approximation might be a good one. This paper is concerned with the sensitivity of these statistical quantities to the underlying statistical structure, when the noise effects are small, and also with the question of when the Gaussian assumption makes sense. Consider a sequence of systems with small noise effects whose statistics converge in some sense to those of a “limit” system. The techniques developed involve approximation and limit theorems for a sequence of variational problems associated with the minimization of the action functionals which arise when the theory of large deviations is applied to the above-mentioned systems. The admissible paths and velocity fields are characterized. Techniques are developed for approximating ε-optimal or optimal paths and values of the action functionals with “restricted velocity fields”, and these are used to get the desired limit, approximation and robustness theorems. Degenerate and nondegenerate cases with both bounded and Gaussian noise are considered. Several examples and an application to a phase locked loop system which arises in communication theory are discussed. These indicate when the Gaussian assumption might be acceptable in practice. The results are of potential use in computation, for they indicate when the results for a simpler “more computable” noise process might be a good approximation to the results for the true noise process. The results concerning convergence and approximation seem to be of independent interest for treating convergence of the solutions of a sequence of more general variational problems. Examples where the “small noise” Gaussian approximation works and does not work are given.