On the parabolic ${\Phi }_3^4$ model for the harmonic oscillator: diagrams and local existence

Abstract

We prove the local wellposedness of the (renormalized) parabolic ${\Phi }_3^4$ model associated with the harmonic oscillator on ${\mathbb{R}}^3$, that is, the equation formally written as \begin{equation*} \partial_t X + HX= -X^3+\infty\cdot X + ξ, \quad t>0, \quad x \in \mathbb{R}^3, \end{equation*} where $H:=-{\Delta }_{{\mathbb{R}}^3}+|x{|}^2$ and $\xi$ denotes a space-time white noise. This model is closely related to the Gross-Pitaevskii equation which is used in the description of Bose-Einstein condensation. Our overall formulation of the problem, based on the so-called paracontrolled calculus, follows the strategy outlined by Mourrat and Weber for the ${\Phi }_3^4$ model on the three-dimensional torus. Significant effort is then required to adapt, within the framework imposed by the harmonic oscillator, the key tools that contribute to the success of this method-particularly the construction of stochastic diagrams at the core of the dynamics.

104 pages