Invariance of white noise for KdV on the line

Abstract

We consider the Korteweg--de Vries equation with white noise initial data, posed on the whole real line, and prove the almost sure existence of solutions. Moreover, we show that the solutions obey the group property and follow a white noise law at all times, past or future. As an offshoot of our methods, we also obtain a new proof of the existence of solutions and the invariance of white noise measure in the torus setting.

References in corpus (11)

• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Invariance of the white noise for KdV
• Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems
• Almost Periodicity in Time of Solutions of the KdV Equation
• Invariant measures and long time behaviour for the Benjamin-Ono equation III
• Nonuniqueness of weak solutions of the nonlinear Schroedinger equation
• Invariant Gibbs Measure for 3D NLW in Infinite Volume
• White noise for KdV and mKdV on the circle
• Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise
• Invariant measure construction at a fixed mass
• Construction of KdV flow I. Tau function via Weyl function

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• The Quasi-Periodic Cauchy Problem for the Generalized Benjamin-Bona-Mahony Equation on the Real Line
• The Cauchy problem for the generalized KdV equation with rough data and random data