The FBSDE approach to sine-Gordon up to $6π$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909986072821760 |
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| author | Gubinelli, Massimiliano Meyer, Sarah-Jean |
| author_facet | Gubinelli, Massimiliano Meyer, Sarah-Jean |
| contents | We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (βφ))_2$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13648 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The FBSDE approach to sine-Gordon up to $6π$ Gubinelli, Massimiliano Meyer, Sarah-Jean Mathematical Physics Probability 81S20, 60H30 We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (βφ))_2$ 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. |
| title | The FBSDE approach to sine-Gordon up to $6π$ |
| topic | Mathematical Physics Probability 81S20, 60H30 |
| url | https://arxiv.org/abs/2401.13648 |