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Fréchet 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.

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Author:Péter Koltai, Han Cheng Lie, Martin Plonka
Document Type:Article
Parent Title (English):Nonlinearity
First Page:4232
Last Page:4257
Year of first publication:2019
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