CS08: New frontiers in stochastic quantisation

date: 7/14/2025, time: 16:00-17:30, room: ICS 139

Organizer: Tommaso Rosati (University of Warwick)

Chair: Tommaso Rosati (University of Warwick)

An introduction to the forward backward SDE for stochastic

Sarah-Jean Meyer (University of Oxford)

In this talk, I will give an present the key ideas behind a new approach to stochastic quantisation \([3]\). It builds on renormalisation group ideas and extends \([1]\). Using a forward backward stochastic differential equation (FBSDE), we are able to construct the sine-Gordon QFT as a scale-by-scale coupling to the Gaussian free field. Our analysis avoids exact solutions to the Polchinsky equation, and instead relies an approximate flow equation similar to \([2]\). In the case of the infinite volume sine-Gordon EQFT up to \(6\pi\), we are able to control the stochastic quantisation equation and we obtain properties like singularity with respect to the free field, exponential decay of correlations, an infinite volume variational problem, the full verification of the Osterwalder Schrader Axiom and non-Gaussianity as a direct consequence. If time allows, I will briefly mention some ongoing work on charactersiations of the sine-Gordon EQFT using a martingale problem and integration by parts formula derived from the FBSDE.

Bibliography

\([1]\) Barashkov, N. and Gubinelli, M. (2020) ‘A variational method for \(\Phi^4_3\)’, Duke Mathematical Journal, 169(17). Available at: https://doi.org/10.1215/00127094-2020-0029.

\([2]\) Duch, P., Gubinelli, M. and Rinaldi, P. (2024) ‘Stochastic quantisation of the fractional \(\Phi^4_3\) model in the full subcritical regime’. arXiv. Available at: https://doi.org/10.48550/arXiv.2303.18112.

\([3]\) Gubinelli, M. and Meyer, S.J. (2024) ‘The FBSDE approach to sine-Gordon up to \(6\pi\)’. arXiv. Available at: https://doi.org/10.48550/arXiv.2401.13648.

Asymptotic Exit Problems for a Singular Stochastic Reaction-Diffusion Equation

Tom Klose (University of Oxford)

We consider a singular stochastic reaction-diffusion equation with a cubic non-linearity on the \(3\)D torus and study its behaviour as it exits a domain of attraction of an asymptotically stable point. Mirroring the results of Freidlin and Wentzell in the finite-dimensional case, we relate the logarithmic asymptotics of its mean exit time and exit place to the minima of the corresponding (quasi-)potential on the boundary of the domain. The challenge, in our setting, is that the stochastic equation is singular such that its solution only lives in a Hölder–Besov space of distributions. The proof accordingly combines a classical strategy with novel controllability statements as well as continuity and locally uniform large deviation results obtained via the theory of regularity structures.

Integration by parts formula and quantum field theory

Francesco Carlo De Vecchi (Università di Pavia)

Characterizing a probability measure through an integration by parts formula is a classical problem in stochastic analysis. It finds applications in (Euclidean) quantum field theory, being related to the solutions of the equations of motion for the correlation functions of the quantum field. We approach this problem in the particular case of quantum field theory with exponential interaction on \(\mathbb{R}^2\), studying a Fokker-Planck-Kolmogorov equation associated to a stochastic quantization equation for such a model. We prove that, under some conditions on the support of the measure, the solution to this Fokker-Planck-Kolmogorov equation exists and is unique, providing a complete characterization of the exponential measure by an integration by parts formula. The talk is based on the joint work \([1]\) with Massimiliano Gubinelli and Mattia Turra.

Bibliography

\([1]\) De Vecchi, Francesco C., Massimiliano Gubinelli, and Mattia Turra. “A singular integration by parts formula for the exponential Euclidean QFT on the plane.” arXiv preprint arXiv:2212.05584 (2022)