  We exploit the relationship between the stochastic Koopman operator and the
Kolmogorov backward equation to construct importance sampling schemes for
stochastic differential equations. Specifically, we propose using
eigenfunctions of the stochastic Koopman operator to approximate the Doob
transform for an observable of interest (e.g., associated with a rare event)
which in turn yields an approximation of the corresponding zero-variance
importance sampling estimator. Our approach is broadly applicable and
systematic, treating non-normal systems, non-gradient systems, and systems with
oscillatory dynamics or rank-deficient noise in a common framework. In
nonlinear settings where the stochastic Koopman eigenfunctions cannot be
derived analytically, we use dynamic mode decomposition (DMD) methods to
compute them numerically, but the framework is agnostic to the particular
numerical method employed. Numerical experiments demonstrate that even coarse
approximations of a few eigenfunctions, where the latter are built from
non-rare trajectories, can produce effective importance sampling schemes for
rare events.
