Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

Abstract

We consider the family of interpolation measures of Gibbs measures and white noise given by $$dQ_{0,\b}^{(p)} = Z_\b^{-1} \ind_{{\int_{\T} u^2\le K\b^{-1/2}\}} e^{-\int_{\T} u^2 +\b \int u^p} dP_{0,\b}$$ 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 straightforward 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 for the modified KdV and the cubic nonlinear Schrödinger equations.

References in corpus (2)

• Invariance of the white noise for KdV
• White noise for KdV and mKdV on the circle

Cited by in corpus (5)

• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• On the one-dimensional cubic nonlinear Schrodinger equation below L^2
• Probabilistic methods for discrete nonlinear Schrödinger equations
• Remarks on nonlinear smoothing under randomization for the periodic KdV and the cubic Szegö equation
• Local and global well-posedness of the stochastic KdV-Burgers equation