Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics

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. *

Computation and Optimal Perturbation of Finite-Time Coherent Sets for Aperiodic Flows Without Trajectory Integration

“…To theoretically justify our approach in the infinite-dimensional setting we need some results from [31] regarding the regularity of the spectrum of \scrP 0,\tau with respect to perturbations of the velocity field. Transferring these results to the regularity of the spectrum of \\bfitG with respect to velocity field perturbations, we derive a first variation of \mu k with respect to u and detail the steps below in section 6.1.…”

“…12], and the noise \varepsi I d\times d is smooth enough to apply the results of [31] to \scrP 0,\tau . Assuming that the singular value \sigma k is simple and isolated, [31, Theorem 5.1] and the paragraph following it guarantee Fr\' echet differentiability of \sigma k and the corresponding singular function with respect to u.…”

“…Instead, a linearized optimization of the eigenvalue of the generator \\bfitG leads to a very simple optimization problem (6.8), which can be solved via a linear system of the same dimension as U . Moreover, the theory holds for infinite-dimensional perturbation spaces U as well, since Fr\' echet differentiability of the transfer operator and its spectrum with respect to perturbing velocity fields has been established in [31].…”

“…In order to use results from[31] we will later make an additional assumption on m.Downloaded 04/15/21 to 44.224.250.200. Redistribution subject to CCBY license © 2020 SIAM.…”

Rigorous Computation of Linear Response for Intermittent Maps

“…Indeed, the work of Ruelle was refined in the uniformly hyperbolic setting [12,25], extended to the partially hyperbolic setting [15], and has been a topic of deep investigation for unimodal maps, see [8], the survey article [7], the recent works [3,9,14,39] and references therein. More recently, the topic of linear response was also studied in the context of random or extended systems [6,16,18,23,31,40,44]. Optimisation of statistichal properties through linear respone was develope in [1,2,22,30].…”