Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator

Coherent sets-regions of phase space that resist mixing-organize transport in stochastic dynamical systems and are encoded in the leading singular functions of the stochastic Koopman operator. These are typically computed numerically and globally; a local, analytic theory near a fixed point has been missing, although fixed points and the transitions between them organize the flow. We supply the local theory for a saddle, the transition state through which probability flows between basins. Naively the construction fails-probability escapes along unstable directions, the invariant measure is non-normalizable, the singular functions leave $L^2$, and singular values exceed one. We show that conditioning on non-escape (a Doob's $h$-transform) repairs this; that the required $h$ is the unique stabilizing solution of an algebraic Riccati equation; and that the resulting slow modes decay at the escape rate along the unstable manifold. We obtain these modes in closed form-products of Hermite polynomials in canonical coordinates times the $h$-function-and use their unitary structure to derive a data-driven estimator: canonical correlation analysis on conditioned trajectories. When a single unstable direction dominates, the leading mode is the reaction coordinate, locally recovering the stable manifold and committor of transition-path theory. The construction connects coherent sets to transition-path theory, to the stabilizing Riccati solution of linear control, and to the time-reversed dynamics of generative diffusion models.