The FBSDE approach to sine-Gordon up to $6\ddot{I}€$

Abstract

We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos ({\hat{I}}^2\ddot{I}†){)}_2$ on the full space up to the second threshold, i.e. for $β^2 < 6 π$. The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition $(X_t{)}_{t⩾0}$ of the interacting Euclidean field $X_{\infty }$ along a scale parameter $t⩾0$. This FBSDE describes the optimiser of the stochastic control representation of the Euclidean QFT introduced by Barashkov and one of the authors. We show that the FBSDE provides a description of the interacting field without cut-offs and that it can be used effectively to study the sine-Gordon measure to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties.

95 pages, reverted the assumption on the perturbation g for Lemma 2.6, minor typographical changes