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The FBSDE approach to sine-Gordon up to $6π$

Massimiliano Gubinelli, Sarah-Jean Meyer

TL;DR

The article develops a rigorous stochastic quantisation framework for the two-dimensional sine-Gordon Euclidean QFT on the full space in the subcritical range $\beta^2<6\pi$, via a scale-decomposed forward-backward SDE that couples a Brownian motion with a scale-interpolated Gaussian free field. It constructs a cut-off-free description by renormalising the Polchinski-type flow and proving well-posedness and stability of the effective FBSDE, yielding convergence to the infinite-volume measure $\nu_{SG}$ as a random shift of the free field $W_\infty$. It then establishes decay of correlations, shows mutual singularity with the Gaussian free field for $\beta^2\ge 4\pi$, and provides a variational representation and large deviations principle in both finite and infinite volume, together with a verification of the Osterwalder--Schrader axioms in the small-coupling regime. The work unifies stochastic control, Polchinski flow analysis, and FBSDE methods to advance the rigorous understanding of the sine-Gordon model on $\mathbb{R}^2$ and its Euclidean QFT properties, including OS compliance and a robust variational description of the Laplace transform and large deviations.

Abstract

We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (β\varphi))_2$ 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 $(X_t)_{t \geqslant 0}$ of the interacting Euclidean field $X_{\infty}$ along a scale parameter $t \geqslant 0$. 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.

The FBSDE approach to sine-Gordon up to $6π$

TL;DR

The article develops a rigorous stochastic quantisation framework for the two-dimensional sine-Gordon Euclidean QFT on the full space in the subcritical range , via a scale-decomposed forward-backward SDE that couples a Brownian motion with a scale-interpolated Gaussian free field. It constructs a cut-off-free description by renormalising the Polchinski-type flow and proving well-posedness and stability of the effective FBSDE, yielding convergence to the infinite-volume measure as a random shift of the free field . It then establishes decay of correlations, shows mutual singularity with the Gaussian free field for , and provides a variational representation and large deviations principle in both finite and infinite volume, together with a verification of the Osterwalder--Schrader axioms in the small-coupling regime. The work unifies stochastic control, Polchinski flow analysis, and FBSDE methods to advance the rigorous understanding of the sine-Gordon model on and its Euclidean QFT properties, including OS compliance and a robust variational description of the Laplace transform and large deviations.

Abstract

We develop a stochastic analysis of the sine-Gordon Euclidean quantum field on the full space up to the second threshold, i.e. for . The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition of the interacting Euclidean field along a scale parameter . 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.
Paper Structure (38 sections, 56 theorems, 469 equations)

This paper contains 38 sections, 56 theorems, 469 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Notation and assumptions
  4. Stochastic control set-up for Gibbs measures
  5. Scale decomposition
  6. The control problem
  7. The effective FBSDE
  8. Analysis of the flow equation
  9. Truncating the flow
  10. The Fourier representation
  11. Estimates on the Fourier coefficients
  12. The renormalised problem
  13. Estimates on the force
  14. Estimates on the potential
  15. Analysis of the FBSDE
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\beta^2 < 6 \pi$. For $\rho \in C^{\infty}_c (\mathbb{R}^2)$ or $\rho \equiv 1$ and $T \in [0, \infty]$, there is scale dependent function $F^{\rho, T} = (F^{\rho, T}_s)_{s \in [0, T]}$ such that up to first order (in $\bar{\lambda}$), $F^{\rho, T}_T$ corresponds perturbatively to the Wick-reno and the associated FBSDE eq:int-FBSDE has a solution $(Z^{\rho, T}, R^{\rho, T}) \in \mathbb{H}^{\i

Theorems & Definitions (117)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3: Boué--Dupuis
  • Lemma 2.4
  • ...and 107 more