The stochastic quantisation of the fractional $\Phi^4$ model in the full subcritical domain

Date: 12 May 2025 - 16 May 2025

I will present a complete proof of stochastic quantisation of a family of subcritical (i.e. superrenormalizable) scalar Euclidean QFT via the flow equation method of Duch. Euclidean QFT are measures on distributional fields which should be considered natural generalisation of Markov processes in higher dimension and which play a fundamental role in the rigorous construction of relativistic quantum fields. Stochastic quantisation is a method to realise such measures as pushforward of Gaussian measures via maps obtained by solving PDEs with random sources. In the last 10 years our understanding of the stochastic quantisation method has progressed greatly giving us new methods to attach the problems of EQFTs. The aim of the minicourse is to present, in most of the details, the various aspects of the construction of a particular class of EQFTs showcasing how probabilistic arguments merge with PDE estimates and renormalization group ideas. If the time permits I will also discuss the many open problems and fundamental issues in our understanding of these and more challenging models.