Global Solutions to Elliptic and Parabolic Φ 4 Models in Euclidean Space

Abstract

We prove the existence of global solutions to singular SPDEs on Rd 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 Φd4 Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the Φd-24 Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.