The Schr\"odinger Ornstein--Uhlenbeck flow: dispersion, restriction and nonlinear dynamics

Abstract

We study the Schr\"odinger evolution generated by the isotropic Ornstein--Uhlenbeck operator \[ \mathscr L=\Delta-\langle x,\nabla\rangle \] in $L^2(\mathbb R^d,d\gamma)$, where $d\gamma$ is the invariant Gaussian measure. Combining the classical lens correspondence between the free Schr\"odinger equation and the harmonic oscillator with Gaussian conjugation, we identify the free Schr\"odinger structure hidden in the Ornstein--Uhlenbeck evolution. The main consequence is a dynamic restriction theorem of spacetime type. After an explicit transformation of spacetime variables, the adjoint Ornstein--Uhlenbeck extension operator becomes an ordinary spacetime Fourier transform restricted to the classical Schr\"odinger paraboloid. On the Ornstein--Uhlenbeck side this corresponds to an explicitly time-dependent Gaussian spacetime measure. Thus a restriction geometry which is neither suggested by the symbol of $\mathscr L$ nor selected by a scaling symmetry of the Ornstein--Uhlenbeck equation emerges dynamically from the free Schr\"odinger representation. The same correspondence identifies the weighted Gaussian spaces in the homogeneous and inhomogeneous Ornstein--Uhlenbeck Strichartz estimates and transports the mass-critical nonlinear Schr\"odinger equation to \[ i\partial_tu+\mathscr Lu = \mu e^{-|x|^2/d}|u|^{4/d}u. \] We also obtain a global description of the unitary Ornstein--Uhlenbeck group, including its oscillatory representation away from the caustics and its exact values at the caustic times, together with the associated sharp weighted dispersive estimate and Hardy-type uncertainty principles.