Title:
Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics
Author(s):
Koltai, Péter; Lie, Han Cheng; Plonka, Martin
Year of publication:
2019
Available Date:
2020-08-19T10:40:19Z
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.
Part of Identifier:
e-ISSN (online): 1361-6544
Keywords:
stochastic differential equations
transfer operator
Koopman operator
Perron–Frobenius operator
smooth drift dependence
linear response
pathwise expectations
Mathematics Subject Classification numbers: 37H99, 47H30, 58C20, 60H07, 60H10
DDC-Classification:
510 Mathematik
Publication Type:
Wissenschaftlicher Artikel
URL of the Original Publication:
DOI of the Original Publication:
Journaltitle:
Nonlinearity
Department/institution:
Mathematik und Informatik