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

I will present some recent results on 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 $d=2,3$. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the ${\mathrm{\Phi }}_d^4$ Euclidean quantum field theory via Parisi--Wu stochastic quantization, while the elliptic equations are linked to the ${\mathrm{\Phi }}_{d-2}^4$ Euclidean quantum field theory via the Parisi--Sourlas dimensional reduction mechanism. We prove existence for the elliptic equations and existence, uniqueness and coming down from infinity for the parabolic equations. Join work with Massimiliano Gubinelli.