Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

Chouk, Khalil; Gubinelli, Massimiliano

doi:10.1080/03605302.2015.1073300

Cited by 39 publications

(54 citation statements)

References 40 publications

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“…constructed (weak controlled) solutions to rough transport equations for vector fields b statisfying a linear growth condition and divb ∈ L ∞ ([0, T ] × R d ) by using rough path theory. Further, it is worth mentioning the paper of Chouk, Gubinelli [11]. Here the authors analyze modulated non-linear Schrödinger equations and improve well-posedness of such equations by means of the irregularity of the modulation.…”

Section: Introductionmentioning

confidence: 99%

Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift

2019

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In this paper we present a new method for the construction of strong solutions of SDE's with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H < 1 2 . Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDE's in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a "local time variational calculus". We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

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Section: Introductionmentioning

confidence: 99%

Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift

2019

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“…It suffices to prove the assertion for S = 0 and T = ∞. By (5.12), for any 16) which implies that (5.2)-(5.4) hold in the time range [0, T 1 ]. Thus, using Theorem 5.1 (i) (see also Remark 5.2) we obtain…”

Section: Strichartz and Local Smoothing Estimatesmentioning

confidence: 93%

“…in the L 2 case is proved in [31] by using the stochastic Strichartz estimates from [10]. In addition, we refer the reader to [22] for results on Schrödinger equations with potential perturbed by a temporal white noise, to [10,11] for results on compact manifolds, and to [16,21,24] for Schrödinger equations with modulated dispersion.In this paper, we are mainly concerned with the asymptotic behavior of solutions to stochastic nonlinear Schrödinger equations. More precisely, we focus on the scattering property of solutions, which is of physical importance and, roughly speaking, means that solutions behave asymptotically like those to linear Schrödinger equations.There is an extensive literature on scattering in the deterministic case.…”

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confidence: 99%

Scattering for Stochastic Nonlinear Schrödinger Equations

Herr

Röckner

Deng

2019

Commun. Math. Phys.

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We study the scattering behavior of global solutions to stochastic nonlinear Schrödinger equations with linear multiplicative noise. In the case where the quadratic variation of the noise is globally finite and the nonlinearity is defocusing, we prove that the solutions scatter at infinity in the pseudo-conformal space and in the energy space respectively, including the energy-critical case. Moreover, in the case where the noise is large, non-conservative and has infinite quadratic variation, we show that the solutions scatter at infinity with high probability for all energy-subcritical exponents.2010 Mathematics Subject Classification. 60H15, 35Q55, 35P25. in the L 2 case is proved in [31] by using the stochastic Strichartz estimates from [10]. In addition, we refer the reader to [22] for results on Schrödinger equations with potential perturbed by a temporal white noise, to [10,11] for results on compact manifolds, and to [16,21,24] for Schrödinger equations with modulated dispersion.In this paper, we are mainly concerned with the asymptotic behavior of solutions to stochastic nonlinear Schrödinger equations. More precisely, we focus on the scattering property of solutions, which is of physical importance and, roughly speaking, means that solutions behave asymptotically like those to linear Schrödinger equations.There is an extensive literature on scattering in the deterministic case. Let α(d) denote the Strauss exponent, see (1.10) below. In the defocusing case, scattering was proved in the pseudo-conformal space (i.e., {u ∈ H 1 :For small initial data, scattering was also obtained for α ∈ (1 + 4/d, 1 + 4/(d − 2)) in [45,15]. Moreover, in the energy space H 1 , the scattering property of solutions was first proved for the inter-critical case where α ∈ (1 + 4/d, 1 + 4/(d − 2)) in [28]. In the much more difficult energy-critical case α = 1 + 4/(d − 2), global well-posedness and scattering are established in [17] for d = 3, in [41] for d = 4, and in [46] for d ≥ 5. Moreover, global well-posedness and scattering in L 2 are proved in [25] for the masscritical exponent α = 1 + 4/d, d ≥ 3. In the focusing case, there exists a threshold for global well-posedness, scattering and blow-up; we refer to [26,33,34,36,37] and references therein.In the framework of stochastic mechanics, developed by E. Nelson [38], there are also several works devoted to potential scattering, in terms of diffusions instead of wave functions. See, e.g., [12,13,43].However, to the best of our knowledge, there are few results on the scattering problem for stochastic nonlinear Schrödinger equations (1.1). One interesting question is that, whether the scattering property is preserved under thermal fluctuations? Furthermore, in the regime where the deterministic system fails to scatter, will the input of some large noise have the effect to improve scattering with high probability?One major challenge here lies in establishing global-in-time Strichartz estimates for (1.1), which actually measure the dispersion and are closely related to those of ti...

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“…Nevertheless, the definition of the transport equation for such irregular vectorfields seems to be tricky in that case. The method of Chouk and Gubinelli [6] and [7] does not apply and another definition has to be found. We postpone the analysis of this situation to a further publication.…”

Section: 1mentioning

confidence: 99%