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

  • Martina Hofmanova (TU Berlin)
A3 01 (Sophus-Lie room)

Abstract

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. Join work with Massimiliano Gubinelli.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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