Interpolation of Gibbs measures with white noise for Hamiltonian PDE

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.