2018/10/03 by Massimiliano Gubinelli, Martina Hofmanová, Gubinelli, Massimiliano +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Complex Systems and Time Series Analysis #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.1810.01700
openalex publication_date 2018/10/03 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
We present a new construction of the Euclidean Φ4 quantum field theory on ℝ3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on ℝ3 defined on a periodic lattice of mesh size ε and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as ε → 0, M → ∞. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder--Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson--Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields.