Uncertainty Propagation in Abstracted Systems via the Liouville Equation

This technical brief shows that given a system and its abstraction, the evolution of uncertain initial conditions in the original system is, in some sense, matched by the evolution of the uncertainty in the abstracted system. In other words, it is shown that the concept of $Φ$-related vector fields extends to the case of stochastic initial conditions where the probability density function (pdf) for the initial conditions is known. In the deterministic case, the $Φ$ mapping commutes with the system dynamics. In this paper, we show that in the case of stochastic initial conditions, the induced mapping $Φpdf$ commutes with the evolution of the pdf according to the Liouville equation.