Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

2010/05/21 by Tadahiro Oh, Jeremy Quastel, Oh, Tadahiro +3 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1005.3957

openalex publication_date 2010/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the family of interpolation measures of Gibbs measures and white noise given by dQ_0,\b^(p) = Z_\b^-1 \ind_∫_\T u²≤ K\b^-1/2\ e^-∫_\T u² +\b ∫ u^p dP_0,\b where P_{0, \b} is the Wiener measure on the circle, with variance β^-1, conditioned to have mean zero. It is shown that as β→ 0, Q₀^β converges weakly to mean zero Gaussian white noise Q₀. As an application, we present a straightforward proof that Q₀ is invariant for the Kortweg-de Vries equation (KdV). This weak convergence also shows that the white noise is a weak limit of invariant measures for the modified KdV and the cubic nonlinear Schrödinger equations.

Citations

• White noise for KdV and mKdV on the circle

Cited by

• Probabilistic methods for discrete nonlinear Schrödinger equations
• Local and global well-posedness of the stochastic KdV-Burgers equation

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