Large Deviations Estimates for Systems with Small Noise Effects, and Applications to Stochastic Systems Theory

摘要

For typical stochastic systems (e.g., tracking systems), estimates of the behavior are hard to get. If the noise effects are small, then asymptotic methods are appealing (to get, for example, estimates of times required to lose track, etc.). In this paper, systems with small noise effects and wide bandwidth noise inputs are analyzed via large deviations methods. Inputs to many systems in control, communications or in physics are not 'white noise', but of certain 'wide bandwidth' types. Since deviations results can be sensitive to the actual noise model used, working with a model that is close to the 'physical' form is important. Several such models are dealt with, where the bandwidth is large but the 'intensity' small. For the models chosen, the action functionals turn out to be the same for the 'small Gaussian white noise' models. Thus, is is actually feasible to do computations with them. Estimates of the probability that the path lies in various sets are obtained. The formula for the mean escape time of the system from a set in which 'average dynamics' are stable is given, as are results on the most likely escape routes and the likely locations of the path on long time intervals. Such quantities, already available with small white noise detailed analysis of a phase locked loop tracking system (a special form of a nonlinear filter). (Author) (Reprints.)