Confidence Domain, Return Time and Control

Abstract

In this chapter we analyse stochastic dynamical systems that have a stable equilibrium. This equilibrium of the corresponding deterministic system represents the “ideal” state of the system. Due to the stochastic perturbations the system fluctuates about this equilibrium state. In order to quantify the functioning of the system we introduce concepts as confidence domain and return time. A confidence domain in state space is a bounded domain containing the equilibrium. Within this domain the probability density function for the stationary distribution is above a fixed value. This value can be selected such that the system is with a given probability within this domain. The return time is the expected time it takes to return to such a domain given a starting point somewhere outside the domain. The approximation for the return time appears to be independent of this starting point. The return time is related to the notion of resilience that is used in ecology, see Sect. 9.2.