A simple construction of the dynamical $\Phi^4_3$ model

TL;DR

This paper introduces a straightforward method to establish the global well-posedness of the singular stochastic PDE known as the $\

Contribution

It presents a simple transformation that reduces the $\

Findings

Provides a new elementary proof of global well-posedness

Eliminates the need for advanced mathematical frameworks

Uses basic estimates and principles for analysis

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.