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We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (\beta \varphi))_2$ on the full space up to the second threshold, i.e. for $\beta^2 < 6 \pi$. 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., Comment: 78 pages

We introduce Wilson-It\^o diffusions, a class of random fields on $\mathbb{R}^d$ that change continuously along a scale parameter via a Markovian dynamics with local coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra \`a la Costello-Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson-Polchinski flow equations., Comment: 8 pages

We present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the H{\o}egh--Krohn (or $\exp (\alpha \phi)_2$) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates while the second on estimates of the Malliavin derivative of the solution to the SQ equation., Comment: 30 pages - accepted version

We introduce a theory of non-commutative $L^{p}$ spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann It\^o processes, Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the $\Phi^{4}_{2}$ Euclidean QFT in a bounded region and weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio and Mitter on the stochastic quantisation of $\Phi^{4}_{2}$., Comment: 59 pages

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a $d$-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in [H. van Beijeren, R. Kutner and H. Spohn, Excess noise for driven diffusive systems, PRL, 1985]. In both the critical $d=2$ and super-critical $d\geq 3$ cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For $d\ge3$ the scaling adopted is the classical diffusive one, while in $d=2$ it is the weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way., Comment: 68 pages, 4 figures. A mistake in an earlier version has been corrected

We present a construction of the fractional $\Phi^4$ Euclidean quantum field theory on $\mathbb{R}^3$ in the full subcritical regime via parabolic stochastic quantisation. Our approach is based on the use of a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the non-linear stochastic partial differential equation describing the interacting field., Comment: 69 pages

We give a novel characterization of the Euclidean quantum field theory with exponential interaction $\nu$ on $\mathbb{R}^2$ through a renormalized integration by parts (IbP) formula, or otherwise said via an Euclidean Dyson-Schwinger equation for expected values of observables. In order to obtain the well-posedness of the singular IbP problem, we import some ideas used to analyse singular SPDEs and we require the measure to "look like" the Gaussian free field (GFF) in the sense that a suitable Wasserstein distance from the GFF is finite. This guarantees the existence of a nice coupling with the GFF which allows to control the renormalized IbP formula., Comment: 78 pages

Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forward-backward stochastic differential equation (FBSDE) which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the so-called continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean fermionic field theories and prove exponential decay of correlations., Comment: The errors in some proofs have been corrected and some typos have been fixed. Accepted for publication in Annals of Probability

We investigate the infinite volume limit of the variational description of Euclidean quantum fields introduced in a previous work. Focussing on two dimensional theories for simplicity, we prove in details how to use the variational approach to obtain tightness of $\varphi^4_2$ without cutoffs and a corresponding large deviation principle for any infinite volume limit. Any infinite volume measure is described via a forward--backwards stochastic differential equation in weak form (wFBSDE). Similar considerations apply to more general $P (\varphi)_2$ theories. We consider also the $\exp (\beta \varphi)_2$ model for $\beta^2 < 8 \pi$ (the so called full $L^1$ regime) and prove uniqueness of the infinite volume limit and a variational characterization of the unique infinite volume measure. The corresponding characterization for $P (\varphi)_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations., Comment: 39 pages, some corrections and remarks on the FBSDE formulation