Global Solutions to Elliptic and Parabolic $${\Phi^4}$$ Φ 4 Models in Euclidean Space

Abstract

We prove the existence of global solutions to singular SPDEs on $${\mathbb{R}^{\rm d}}$$ 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.