Koopman operators linearize nonlinear dynamical systems, making their
spectral information of crucial interest. Numerous algorithms have been
developed to approximate these spectral properties, and Dynamic Mode
Decomposition (DMD) stands out as the poster child of projection-based methods.
Although the Koopman operator itself is linear, the fact that it acts in an
infinite-dimensional space of observables poses various challenges. These
include spurious modes, essential spectra, and the verification of Koopman mode
decompositions. While recent work has addressed these challenges for
deterministic systems, there remains a notable gap in verified DMD methods
tailored for stochastic systems, where the Koopman operator measures the
expectation of observables. We show that it is necessary to go beyond
expectations to address these issues. By incorporating variance into the
Koopman framework, we address these challenges. Through an additional DMD-type
matrix, we approximate the sum of a squared residual and a variance term, each
of which can be approximated individually using batched snapshot data. This
allows verified computation of the spectral properties of stochastic Koopman
operators, controlling the projection error. We also introduce the concept of
variance-pseudospectra to gauge statistical coherency. Finally, we present a
suite of convergence results for the spectral quantities of stochastic Koopman
operators. Our study concludes with practical applications using both simulated
and experimental data. In neural recordings from awake mice, we demonstrate how
variance-pseudospectra can reveal physiologically significant information
unavailable to standard expectation-based dynamical models