An introduction to the FBSDE approach for the stochastic quantisation of sine-Gordon

I will introduce a novel approach to constructing Euclidean quantum field theories defined as Gibbs measures on the space of Schwartz distributions. The method is based on a forward-backward stochastic differential equation (FBSDE), which encodes the dynamics of moving between scales and provides a continuous, probabilistic implementation of the continuous renormalisation group (RG), without requiring exact solutions to the RG flow equation.

This framework offers tight control over the measure, enabling the derivation of several structural properties: a variational formulation and large deviation principle, exponential decay of correlations, a characterisation of the singularity relative to the free field, a proof of non-Gaussianity and a complete verification of the Osterwalder–Schrader axioms.

The talk is based on arXiv:2401.13648, joint work with Massimiliano Gubinelli.