Interpolation of Gibbs measures with White Noise for Hamiltonian PDE

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We consider the family of interpolation measures of Gibbs measures and white noise given by dQ₀,⁽ᵖ⁾ = Z⁻¹ ∫ u²≤ K⁻¹/²} e⁻∫ ᵘ² ⁺ ∫ ᵘᵖ dP₀, where P₀, is the Wiener measure on the circle, with variance β⁻¹, conditioned to have mean zero. It is shown that as β→ 0, Q₀β 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.