Work

Invariance of Gaussian RKHSs Under Koopman Operators of Stochastic Differential Equations With Constant Matrix Coefficients

Year: 2024

Type: article

Abstract: ABSTRACT We consider the Koopman operator semigroup associated with stochastic differential equations of the form with constant matrices and and Brownian motion . We prove that the reproducing kernel Hilbert space generated by a Gaussian kernel with a positive definite covariance matrix is invariant under each Koopman operator if the matrices , , and satisfy the following Lyapunov‐like matrix inequality: . In this course, we prove a characterization concerning the inclusion of Gaussian RKHSs for two positive definite matrices and . The question of whether the sufficient Lyapunov‐condition is also necessary is left as an open problem.

Source: PAMM

DOI: 10.1002/pamm.202400127

Authors: Friedrich M. Philipp, Manuel Schaller, Karl Worthmann, Sebastian Peitz, Feliks Nüske

Institutions: Technische Universität Ilmenau, Paderborn University, Max Planck Institute for Dynamics of Complex Technical Systems, Freie Universität Berlin

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: Carl-Zeiss-Stiftung, Deutsche Forschungsgemeinschaft, Bundesministerium für Bildung und Forschung