The Distributional Koopman Operator for Random Dynamical Systems

The Distributional Koopman Operator (DKO) is introduced as a way to perform Koopman analysis on random dynamical systems where only aggregate distribution data is available, thereby eliminating the need for particle tracking or detailed trajectory data. Our DKO generalizes the stochastic Koopman operator (SKO) to allow for observables of probability distributions, using the transfer operator to propagate these probability distributions forward in time. Like the SKO, the DKO is linear with semigroup properties, and we show that the dynamical mode decomposition (DMD) approximation can converge to the DKO in the large data limit. The DKO is particularly useful for random dynamical systems where trajectory information is unavailable.