Robust Approximation of the Stochastic Koopman Operator

We analyze the performance of Dynamic Mode Decomposition (DMD)-based approximations of the stochastic Koopman operator for random dynamical systems where either the dynamics or observables are affected by noise. Under certain ergodicity assumptions, we show that standard DMD algorithms converge provided the observables do not contain any noise and span an invariant subspace of the stochastic Koopman operator. For observables with noise, we introduce a new, robust DMD algorithm that can approximate the stochastic Koopman operator and demonstrate how this algorithm can be applied to Krylov subspace based methods using a single observable measured over a single trajectory. We test the performance of the algorithms over several examples.