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21
Invariance of the Gibbs measure for the SchrödingerBenjaminOno system
 SIAM J. MATH. ANAL
"... We prove the invariance of the Gibbs measure for the periodic SchrödingerBenjaminOno system (when the coupling parameter γ 6 = 0, 1) by establishing a new local wellposedness in a modified Sobolev space and constructing the Gibbs measure (which is in the subL² setting for the BenjaminOno par ..."
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Cited by 19 (6 self)
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We prove the invariance of the Gibbs measure for the periodic SchrödingerBenjaminOno system (when the coupling parameter γ 6 = 0, 1) by establishing a new local wellposedness in a modified Sobolev space and constructing the Gibbs measure (which is in the subL² setting for the BenjaminOno part.) We also show the illposedness result in Hs(T) × Hs− 1 2 (T) for s < 1 2 when γ  6 = 0, 1 and for any s ∈ R when γ  = 1.
Invariant weighted Wiener measures and almost sure global wellposedness for the periodic derivative
 NLS, J. Eur. Math. Soc. (JEMS
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Interpolation of Gibbs measures with White Noise for Hamiltonian PDE, arXiv:1005.3957v1 [math.PR
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Modulation spaces, Wiener amalgam spaces, and Brownian motions
 Adv. Math
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Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation, preprint
"... Abstract. We consider Cauchy problems of dispersive PDEs with random initial data. In particular, we construct localintime solutions to the meanzero periodic KdV almost surely for the initial data in the support of the meanzero Gaussian measures on Hs(T), s> s0, for some s0 < − 1 2, by exh ..."
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Cited by 10 (4 self)
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Abstract. We consider Cauchy problems of dispersive PDEs with random initial data. In particular, we construct localintime solutions to the meanzero periodic KdV almost surely for the initial data in the support of the meanzero Gaussian measures on Hs(T), s> s0, for some s0 < − 1 2, by exhibiting nonlinear smoothing under randomization on the second iteration of the integration formulation. We also show that there is no nonlinear smoothing for the dispersionless cubic Szego ̈ equation under randomization of initial data. 1.
Random data Cauchy theory for nonlinear wave equations of powertype on R3, arXiv:1309.1225 [math.AP
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ALMOST SURE WELLPOSEDNESS OF THE CUBIC NONLINEAR SCHRÖDINGER EQUATION BELOW L²(T)
, 2011
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DIOPHANTINE CONDITIONS IN WELLPOSEDNESS THEORY OF COUPLED KDVTYPE SYSTEMS: LOCAL THEORY
, 2009
"... We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we u ..."
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Cited by 5 (2 self)
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We consider the local wellposedness problem of a oneparameter family of coupled KdVtype systems both in the periodic and nonperiodic setting. In particular, we show that certain resonances occur, closely depending on the value of a coupling parameter α when α = 1. In the periodic setting, we use the Diophantine conditions to characterize the resonances, and establish sharp local wellposedness of the system in H s (Tλ), s ≥ s ∗, where s ∗ = s ∗ (α) ∈ ( 1,1] is determined by the Diophantine characterization of certain 2 constants derived from the coupling parameter α. We also present a sharp local (and global) result in L 2 (R). In the appendix, we briefly discuss the local wellposedness result in H −1 2 (Tλ) for α = 1 without the mean 0 assumption, by introducing the vectorvalued X s,b spaces.
White noise for KdV and mKdV on the circle
"... Abstract. We survey different approaches to study the invariance of the white noise for the periodic KdV. We mainly discuss the following two methods. First, we discuss the PDE method, following Bourgain [2], in a general framework. Then, we show how it can be applied to the low regularity setting o ..."
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Cited by 3 (2 self)
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Abstract. We survey different approaches to study the invariance of the white noise for the periodic KdV. We mainly discuss the following two methods. First, we discuss the PDE method, following Bourgain [2], in a general framework. Then, we show how it can be applied to the low regularity setting of the white noise for KdV by introducing the Besovtype space b̂sp,∞, sp < −1. Secondly, we describe the probabilistic method by Quastel, Valkó, and the author [41]. We also use this probabilistic approach to study the white noise for mKdV. 1.