Interpolation of Gibbs measures with white noise for Hamiltonian PDE

Tadahiro Oh, Jeremy Quastel*, Benedek Valko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the family of interpolation measures of Gibbs measures and white noise given by measures $dQ_{0,\b}^{(p)} = Z_\b^{-1} \ind_{\{\int u^2\le K\b^{-1/2}\}} e^{ -\int u^2 +\b \int u^p } dP_{0,\b}$ where $P_{0, \b}$ is the Wiener measure on the circle, with variance $\beta^{-1}$, conditioned to have mean zero. It is shown that
as $\beta\to 0$, $Q_0^\beta$ converges weakly to mean zero Gaussian white noise $Q_0$. As an application, we present  a reasonably direct proof that $Q_0$ 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 under the flow for the modifier KdV and the cubic nonlinear Schr\"odinger equation.

Original language English 391-410 20 Journal de Mathématiques Pures et Appliquées 97 4 https://doi.org/10.1016/j.matpur.2011.11.003 Published - Apr 2012

Keywords

• white noise
• Gibbs measure
• Kortweg-de Vries equation
• Schrodinger equation