Via Zoom: A Koopman Framework for Sampling in Stochastic Differential Equations

We propose a general framework to construct efficient sampling methods for stochastic differential equations (SDEs) using eigenfunctions of the system’s Koopman operator. Importance sampling for SDEs is typically done by adding a control term in the drift so that the resulting estimator has a lower variance. The optimal choice for the control term is given by the Doob transform, in which the resulting estimator has zero variance. The Koopman operator is a tool from the dynamical systems community that has recently gained much interest in data-driven analysis of dynamical systems. We show how eigenfunctions of an SDE’s Koopman operator can be used to construct approximations of the Doob transform.