I will present a new approach to stochastic quantisation using a forward-backward stochastic differential equation (FBSDE) for a decomposition X_t of the interacting Euclidean quantum field theory (EQFT) X_∞ along a scale parameter t≥0. This FBSDE provides a pathwise scale-by-scale coupling of the Gaussian free field and the interacting EQFT and is related to the stochastic control representation for EQFTs introduced by Barashkov-Gubinelli.
In the case of the sine-Gordon EQFT on the full space, we show that in the first few regions of collapse, this FBSDE provides a description of the interacting field without cut-offs and that it can be used effectively to study the sine-Gordon measure and obtain results such as large deviations, integrability, decay of correlations for local observables, singularity and a verification of the Osterwalder-Schrader axioms.
This is joint work with Massimiliano Gubinelli.