Global Well-posedness and Scattering for Stochastic generalized KdV Equations with additive noise

2025/11/10 by Engin Başakoğlu, Başakoğlu, Engin, Faruk Temur +3
Mathematics · Economics, Econometrics and Finance · #Advanced Mathematical Physics Problems #Stochastic processes and financial applications #Gas Dynamics and Kinetic Theory

paper · pdf · doi:10.48550/arxiv.2511.07386

Abstract

We study the defocusing stochastic generalized Korteweg-de Vries equations (sgKdV) driven by additive noise, with a focus on mass-critical and supercritical nonlinearities. For integers k ≥ 4, we establish local well-posedness almost surely up to scaling critical regularity. We also prove global well-posedness and scattering in L²_x(ℝ) for the mass-critical equation with small initial data; also in H¹_x(ℝ) for the mass supercritical equation. In particular, we prove oscillatory integral estimates associated with more general dispersion relations, which are of independent interest; and we make use of a special case of these estimates as a main ingredient for the necessary bounds on the tail of the stochastic convolution for sgKdV, which is crucial to conclude scattering results.

Citations

• Scattering for Stochastic Nonlinear Schrödinger Equations with additive noise
• On long time behavior for stochastic nonlinear Schrödinger equations with a multiplicative noise
• Almost conservation laws for stochastic nonlinear Schrödinger equations
• Well-posedness of distribution dependent SDEs with singular drifts
• Global well-posedness and scattering for the defocusing, mass - critical\n generalized KdV equation
• Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation
• Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise
• Almost conservation laws and global rough solutions to a Nonlinear Schrödinger equation

Related

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• Stochastic nonlinear Schroedinger equations in the defocusing mass and energy critical cases
• Stochastic Schrödinger-Korteweg de Vries systems driven by multiplicative noises