  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 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 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 information 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.
