Interpolation of Gibbs measures with white noise for Hamiltonian PDE (Q409654)

Consider the circle \(\mathbf{T}=\mathbb R/\mathbb Z\), and let \(Q_0\) denote the Gaussian white noise measure defined on tempered distributions \(u\in \mathcal S'(\mathbf{T})\) satisfying \(\int e^{i\langle u,f\rangle}dQ_0(u)=e^{-\frac12\|f\|^2_{L^2}}\) for \(f\in\mathcal S(\mathbf{T}).\) Let \(P_{0,\beta}\) denote the Wiener measure with zero mean and variance \(\beta^{-1}\) on \(C(\mathbf{T})\).NEWLINENEWLINEThe authors consider the family of interpolation measures given by NEWLINE\[NEWLINEdQ_{0,\beta}^{(p)}=Z_\beta^{-1}\mathbf{1}_{\{ \int_\mathbf{T} u^2\leq K\beta^{-\frac12} \}}e^{-\frac12\int_\mathbf{T} u^2+\beta\int_\mathbf{T} u^p} \;dP_{0,\beta},NEWLINE\]NEWLINE where \(Z_\beta^{-1}\) is a normalizing constant to obtain a probability measure and \(p\) describes the order of the nonlinearity. It is shown that as \(\beta\rightarrow 0\), \(Q_{0,\beta}^{(p)}\) converges weakly to the white noise measure \(Q_0\). As an application, the authors prove that \(Q_0\) is invariant for the Korteweg-de Vries equation (KdV). This weak convergence also implies that \(Q_0\) is a weak limit of invariant measures for the modified Korteweg-de Vries equation (mKdV) and the cubic nonlinear Schrödinger equation (NLS).