Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift

Flandoli, Franco; Gubinelli, Massimiliano; Priola, Enrico

doi:10.1016/j.bulsci.2010.02.003

Cited by 52 publications

(71 citation statements)

References 30 publications

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“…Gubinelli and Priola [13]). Moreover, the inverse flow Y s,t (x) := X −1 s,t (x) satisfies the following backward stochastic differential equation…”

Section: Notationsmentioning

confidence: 94%

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$W^{1,p}$-Solutions of the transport equation by stochastic perturbation

Mollinedo¹

2020

Braz. J. Probab. Stat.

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“…Gubinelli and Priola [13]). Moreover, the inverse flow Y s,t (x) := X −1 s,t (x) satisfies the following backward stochastic differential equation…”

Section: Notationsmentioning

confidence: 94%

“…From Flandoli, Gubinelli and Priola [13,Theorem 7], we also remember the following result that we are going to use in our main results: Let b n ∈ C θ (R d , R d ) and let φ n be the corresponding stochastic flows. Assume that…”

Section: Notationsmentioning

confidence: 99%

$W^{1,p}$-Solutions of the transport equation by stochastic perturbation

Mollinedo¹

2020

Braz. J. Probab. Stat.

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“…The same dependencies also arise for non-deterministic systems, e.g. [FGP10,AJKW17] for Fréchet-type dependencies on the initial data, and [FLL + 99, GM05, Mon13,DG14] for the dependence of path functionals or their expectations with respect to changes in the drift. This paper uses the approach of [FLL + 99], which establishes a Gâteaux-type dependence on the data by establishing the existence of directional derivatives with respect to the drift, in order to establish the Fréchet-type dependence of the solution operator with respect to an additive change of drift in a sufficiently smooth setting: for a suitable observable g, we provide in Theorem 3.1 the Fréchet derivative at γ = 0 of the non-linear functional u x g (γ) := E g(X γ ) X γ 0 = x with respect to additive perturbations in γ; above, X γ denotes the solution of the perturbed stochastic differential equation (2.7) below.…”

Section: Introductionmentioning

confidence: 92%

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

2019

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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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“…satisfy regularity conditions that guarantee the existence and the uniqueness of mild solutions, as well as the continuity of the trajectories; see, e.g. [Cer01,FGP10] for such conditions in the case of locally Lipschitz coefficients. It is well-known that the evolution of the probability density of the random R p -valued variable, X t (solving Eq.…”

Section: Ruelle-pollicott Resonances and The Decomposition Of Correlamentioning

confidence: 99%

Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation

et al. 2019

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The response of a low-frequency mode of climate variability, El Niño-Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane-Zebiak model [ZC87], from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum.Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle-Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution [CTND19] to model simulations.The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by [Gas02] and made explicit in the second part of this contribution [TCND19]. Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case.More generally, the reduction approach theorized in [CTND19], complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems. KeywordsRuelle-Pollicott resonances · Stochastic Bifurcation · Markov matrix · ENSO IntroductionComplex and unpredictable behavior of trajectories is observed for many physical systems. This can be due to interactions with many degrees of freedom which can be modeled by a stochastic forcing or to nonlinear coupling resulting in chaotic trajectories. As a result, prediction beyond a certain horizon is hopeless and one focuses instead on the statistical evolution of the system. This loss of predictability manifests itself by the evolution of an ensemble of trajectories becoming independent on its initial condition after a given time, the mixing time. This notion of mixing in state space is in turn closely linked to the correlation function of a pair of observables, which assigns to any positive time lag the correlation between the first observable and the lagged version of the second, and thus gives a measure of the statistical dependence A. Tantet

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Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift

Cited by 52 publications

References 30 publications

$W^{1,p}$-Solutions of the transport equation by stochastic perturbation

$W^{1,p}$-Solutions of the transport equation by stochastic perturbation

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

Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation

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