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## Global solutions to elliptic and parabolic $\Phi^4$ models in Euclidean space Hofmanova, Martina | CIRM H

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Research talks;Partial Differential Equations;Mathematical Physics;Probability and Statistics

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 $\Phi^4_d$ 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. We prove existence for the elliptic equations and existence, uniqueness and coming down from infinity for the parabolic
equations. Joint work with Massimiliano Gubinelli.
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 $\Phi^4_d$ Euclidean quantum field ...

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