Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space

TL;DR

This paper establishes the existence and uniqueness of global solutions to singular stochastic PDEs with cubic nonlinearities in Euclidean space, connecting to quantum field theories through stochastic quantization and dimensional reduction.

Contribution

It provides the first rigorous proof of global solutions for elliptic and parabolic $ ext{Phi}^4$ models in multiple dimensions, advancing the mathematical understanding of quantum field theories.

Findings

Proves existence of solutions in elliptic and parabolic cases for specific dimensions.

Establishes uniqueness and 'coming down from infinity' for parabolic equations.

Links stochastic PDE solutions to Euclidean quantum field theories via established mechanisms.

Abstract

We prove existence of global solutions to singular SPDEs on $\mathbb{R}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $\Phi^4_d$ Euclidean quantum field theory via Parisi--Wu stochastic quantization, while the elliptic equations are linked to the $\Phi^4_{d-2}$ Euclidean quantum field theory via the Parisi--Sourlas dimensional reduction mechanism.