# [[PROB Seminar jointly with NLPDE Seminar]] A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory

Abstract: We present a self-contained construction of the Euclidean $\Phi^4$ quantum field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We introduce an energy method and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. We show that every limit point satisfies reflection positivity, translation invariance and nontriviality (i.e. non-Gaussianity). Our argument applies to arbitrary positive coupling constant and also to multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano Gubinelli.