Mathematics

We present the relationship between quantum field theory (specifically, in the Weightman/Osterwalder-Schrader formulation) and singular Gibbs measures on spaces of distributions. Our discussion involves an overview of various stochastic methods used to construct such measures, with a particular emphasis on characterizing these Gibbs measures through an integration by parts formula (closely related with the Schwinger-Dyson equations for n-point functions). In the case of a scalar bosonic field with exponential interaction, we prove the existence and uniqueness of the measure satisfying the aforementioned integration by parts formula. This presentation is based on a collaboration with Massimiliano Gubinelli and Mattia Turra.