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arXiv · 2609.12026

On the parabolic $Φ_3^4$ model for the harmonic oscillator II: global existence and invariant measures

Aurélien Deya·Reika Fukuizumi·Laurent Thomann
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Abstract

We establish an a priori bound for the dynamical parabolic $Φ_3^4$ 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 $Φ_3^4$ 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.

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Aurélien Deya, Reika Fukuizumi, Laurent Thomann. 2026-09-10. On the parabolic $Φ_3^4$ model for the harmonic oscillator II: global existence and invariant measures. https://arxiv.org/abs/2609.12026

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