Leakage Rate as a Measure of Continuous-Time Stochastic Set Invariance

Probabilistic set invariance is a concept that measures at which probability the given set is invariant under a stochastic process. While this concept successfully measures the level of invariance of a given set, its major limitations include excessive conservativeness and lack of consideration of the prior distribution of the state. In this paper, we introduce a modified notion of probabilistic set invariance that takes into account the starting distribution of the stochastic process. The probability of never escaping the given set, i.e., the survival rate, is defined as a function of the initial distribution and the length of the time window, and its infinitesimal-time behavior is analyzed. For Itô diffusion processes, we find that the decaying rate of survival rate, which we call the leakage rate, can be analytically evaluated through a surface integral formula on the boundary of the set given the probability distribution of state and the update rule. We validate the formula through a numerical example, in which the simulation result well matches the analytical prediction.