Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics. (30th September 2019)

• Record Type: Journal Article
• Authors: Koltai, Péter; Lie, Han Cheng; Plonka, Martin
• Is Part Of: Nonlinearity. Volume 32:Number 11(2019)
• Page Start: 4232
• Page End: 4257
• Publication Date: 2019-09-30
• Subjects: stochastic differential equations -- transfer operator -- Koopman operator -- Perron–Frobenius operator -- smooth drift dependence -- linear response -- pathwise expectations; 37H99 -- 47H30 -- 58C20 -- 60H07 -- 60H10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab1f2a
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. 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.