On the parabolic $Φ_3^4$ model for the harmonic oscillator: diagrams and local existence
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| Format: | Preprint |
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2025
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| _version_ | 1866913776135045120 |
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| author | Deya, Aurélien Fukuizumi, Reika Thomann, Laurent |
| author_facet | Deya, Aurélien Fukuizumi, Reika Thomann, Laurent |
| contents | We prove the local wellposedness of the (renormalized) parabolic $Φ^4_3$ 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 $H:=-Δ_{\mathbb{R}^3} +|x|^2$ and $ξ$ 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 $Φ^4_3$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the parabolic $Φ_3^4$ model for the harmonic oscillator: diagrams and local existence Deya, Aurélien Fukuizumi, Reika Thomann, Laurent Probability Analysis of PDEs 60H15, 35K15, 35R60 We prove the local wellposedness of the (renormalized) parabolic $Φ^4_3$ 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 $H:=-Δ_{\mathbb{R}^3} +|x|^2$ and $ξ$ 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 $Φ^4_3$ 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. |
| title | On the parabolic $Φ_3^4$ model for the harmonic oscillator: diagrams and local existence |
| topic | Probability Analysis of PDEs 60H15, 35K15, 35R60 |
| url | https://arxiv.org/abs/2504.03232 |