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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.
Highlights
In the study of dynamical systems and differential equations in particular, one important aspect is the sensitivity of the solution with respect to the data
We prove the Fréchet differentiability with respect to the drift of Perron– Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations
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
Summary
In the study of dynamical systems and differential equations in particular, one important aspect is the sensitivity of the solution with respect to the data. Results for non-deterministic systems arise in the context of random compositions of maps [BRS17], or for stochastic ordinary and partial differential equations in the weak topology [HM10] The latter reference shows Gâteaux-type pointwise differentiability of the transfer operators acting on smooth functions with respect to a real parameter, where the (constant-in-space) diffusion matrix can vary with the parameter. We collect some relevant results on stochastic processes and transition kernels in the appendices
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