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PreprintCC BY 4.0HAL

On the parabolic Φ34 model for the harmonic oscillator: diagrams and local existence

  • Aurélien Deya
  • Reika Fukuizumi
  • Laurent Thomann
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Abstract

We prove the local well-posedness of the (renormalized) parabolic Φ43 model associated with the harmonic oscillator on ℝ3, that is, the equation formally written as ∂t X + HX= −X3+∞· X + ξ, t>0, x ∈ ℝ3, where H:=−Δℝ3 +|x|2 and ξ denotes a space-time white noise. This model is related to the stochastic Gross-Pitaevskii equation, used in the description of Bose-Einstein condensation. \smallskip Our overall formulation of the problem, based on the so-called paracontrolled calculus, follows the strategy outlined by Mourrat and Weber for the Φ43(Δ) model on the three-dimensional torus. To extend the paracontrolled approach to the harmonic oscillator setting, we develop several new analytical tools. These include a microlocal analysis adapted to the paracontrolled framework, resonance-type/high–low frequency operators, and the construction of the stochastic diagrams governing the dynamics. Taken together, these developments constitute the core technical contribution of the present work.

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Cite

BibTeXRIS
BibTeX
@misc{deya2026parabolic,
  title = {{On the parabolic $\Phi_3^4$ model for the harmonic oscillator: diagrams and local existence}},
  author = {Aurélien Deya and Reika Fukuizumi and Laurent Thomann},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05016338v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the parabolic Φ_3^4 model for the harmonic oscillator: diagrams and local existence
AU  - Aurélien Deya
AU  - Reika Fukuizumi
AU  - Laurent Thomann
PY  - 2026
DA  - 2026/09/01
PB  - HAL
UR  - https://hal.science/hal-05016338v2
AB  - We prove the local well-posedness of the (renormalized) parabolic Φ^4_3 model associated with the harmonic oscillator on ℝ^3, that is, the equation formally written as ∂_t X + HX= −X^3+∞· X + ξ, t>0, x ∈ ℝ^3, where H:=−Δ_ℝ^3 +|x|^2 and ξ denotes a space-time white noise. This model is related to the stochastic Gross-Pitaevskii equation, used in the description of Bose-Einstein condensation. \smallskip 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(Δ) model on the three-dimensional torus. To extend the paracontrolled approach to the harmonic oscillator setting, we develop several new analytical tools. These include a microlocal analysis adapted to the paracontrolled framework, resonance-type/high–low frequency operators, and the construction of the stochastic diagrams governing the dynamics. Taken together, these developments constitute the core technical contribution of the present work.
ER  -