On the parabolic ${\Phi }_3^4$ model for the harmonic oscillator II: global existence and invariant measures

Abstract

We establish an a priori bound for the dynamical parabolic ${\Phi }_3^4$ model with harmonic potential. This bound yields the global well-posedness of the equation and, via the Krylov-Bogoliubov method, the existence of an invariant measure, shown to be non-Gaussian. The argument builds on the strategy developed by Mourrat and Weber for the periodic ${\Phi }_3^4$ model, with substantial modifications to handle the non-compact geometry of ${\mathbb{R}}^3$ and the spectral framework imposed by the harmonic oscillator. We further prove that this measure is unique in the small-coupling regime.

132 pages