Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

Details Interpolation of Gibbs measures with White Noise for Hamiltonian PDE Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

2010-05-21

arxiv.org/abs/1005.3957v1

Mathematics

Probability

Scientific paper

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 Schr\"odinger equations.

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