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Frechet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics
- 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.
| Author details: | Peter KoltaiORCiDGND, Han Cheng LieORCiD, Martin Plonka |
|---|---|
| DOI: | https://doi.org/10.1088/1361-6544/ab1f2a |
| ISSN: | 0951-7715 |
| ISSN: | 1361-6544 |
| Title of parent work (English): | Nonlinearity |
| Publisher: | IOP Publ. Ltd. |
| Place of publishing: | Bristol |
| Publication type: | Article |
| Language: | English |
| Date of first publication: | 2019/09/30 |
| Publication year: | 2019 |
| Release date: | 2020/10/06 |
| Tag: | Koopman operator; Perron-Frobenius operator; linear response; pathwise expectations; smooth drift dependence; stochastic differential equations; transfer operator |
| Volume: | 32 |
| Issue: | 11 |
| Number of pages: | 26 |
| First page: | 4232 |
| Last Page: | 4257 |
| Funding institution: | Deutsche Forschungsgemeinschaft (DFG)German Research Foundation (DFG) [CRC 1114]; Einstein Foundation Berlin (Einstein Center ECMath); Freie Universitat Berlin within the Excellence Initiative of the DFG; open topic postdoctoral fellowship of Universitat Potsdam; DFGGerman Research Foundation (DFG) [SPP 1881] |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
| DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Peer review: | Referiert |
| Publishing method: | Open Access |
| Open Access / Hybrid Open-Access |
