Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator
Abstract
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 L2, 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.
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BibTeX
@misc{ding2026local,
title = {{Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator}},
author = {Xuehao Ding},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35627},
eprint = {2610.35627},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35627},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Koopman modes of saddle-point diffusions: exact singular functions and a data-driven estimator
AU - Xuehao Ding
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35627
UR - https://arxiv.org/abs/2610.35627
AB - 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.
ER -