A simple construction of the dynamical Φ⁴₃ model

Jagannath, Aukosh; Perkowski, Nicolas

doi:10.1090/tran/8724

A simple construction of the dynamical $\Phi ^4_3$ model
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by Aukosh Jagannath and Nicolas Perkowski;

Trans. Amer. Math. Soc. 376 (2023), 1507-1522

DOI: https://doi.org/10.1090/tran/8724

Published electronically: January 4, 2023

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Abstract:

The $\Phi ^4_3$ equation is a singular stochastic PDE with important applications in mathematical physics. Its solution usually requires advanced mathematical theories like regularity structures or paracontrolled distributions, and even local well-posedness is highly nontrivial. Here we propose a multiplicative transformation to reduce the periodic $\Phi ^4_3$ equation to a well-posed random PDE. This leads to a simple and elementary proof of global well-posedness, which only relies on Schauder estimates, the maximum principle, and basic estimates for paraproducts, and in particular does not need regularity structures or paracontrolled distributions.

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