Fr\'echet differentiable drift dependence of Perron--Frobenius and Koopman operators for non-deterministic dynamics

TL;DR

This paper proves that Perron-Frobenius and Koopman operators for stochastic differential equations depend smoothly on the drift, enabling better understanding and control of non-deterministic dynamical systems.

Contribution

It establishes Fréchet differentiability of these operators with respect to the drift, a novel result linking stochastic analysis and operator theory.

Findings

Differentiability of pathwise expectations via Girsanov's formula

Continuous differentiability of eigenvalues and eigenfunctions

Enhanced understanding of operator dependence on system parameters

Abstract

We consider Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations, and establish their Fr\'{e}chet differentiability with respect to the drift. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron-Frobenius and Koopman operators.