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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 van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. ). In both the critical d = 2 and super-critical d ≥ 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 ≥ 3 the scaling adopted is the classical diffusive one, while in d = 2 it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way. [ABSTRACT FROM AUTHOR]

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed here from the point of view of quantum probability: a Grassmann random variable is an homomorphism of an abstract Grassmann algebra into a quantum probability space, i.e. a C ∗ -algebra endowed with a suitable state. We define the notion of Gaussian processes, Brownian motion and stochastic (partial) differential equations taking values in Grassmann algebras. We use them to study the long time behavior of finite and infinite dimensional Langevin Grassmann stochastic differential equations driven by Gaussian space-time white noise and to describe their invariant measures. As an application we give a proof of the stochastic quantization and of the removal of the space cut-off for the Euclidean Yukawa model. [ABSTRACT FROM AUTHOR]

We present a new construction of the Euclidean Φ 4 quantum field theory on R 3 based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on R 3 defined on a periodic lattice of mesh size ε and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as ε → 0 , M → ∞ . Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields. [ABSTRACT FROM AUTHOR]

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for "energy solutions" of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic. [ABSTRACT FROM AUTHOR]

We prove the existence of global solutions to singular SPDEs on R d with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions d = 4, 5 and in the parabolic setting for d = 2, 3. We prove uniqueness and coming down from infinity for the parabolic equations. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the Φ d 4 Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the Φ d - 2 4 Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism. [ABSTRACT FROM AUTHOR]

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Gonçalves and Jara (Arch Ration Mech Anal 212(2):597-644, 2014) and the corresponding uniqueness result of Gubinelli and Perkowski (Energy solutions of KPZ are unique, 2015). [ABSTRACT FROM AUTHOR]

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between the KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also show that there is a natural approximation scheme for the nonlinearity in the stochastic Burgers equation. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation that does not rely on the Cole-Hopf transform. [ABSTRACT FROM AUTHOR]

We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable Hamiltonian functions. The approximate models we analyze (essentially discrete and continuous vortex filaments and vortex loops) are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high Reynolds number. Finally, we apply our strategy to discrete models for filaments used in numerical methods. [ABSTRACT FROM AUTHOR]

In this note, we study the non-linear evolution problem where $X$ is a $\gamma$ -Hölder continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $\gamma$ in $(0,1)$ ), and then when $B$ satisfies some Lipschitz type conditions (for $\gamma>1/2$ ). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type. [ABSTRACT FROM AUTHOR]

We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known. In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include: [ABSTRACT FROM AUTHOR]

We present a Monte Carlo study of the high-temperature phase of the two-dimensional driven lattice gas at infinite driving field. We define a finite-volume correlation length, verify that this definition has a good infinite-volume limit independent of the lattice geometry, and study its finite-size-scaling behavior. The results for the correlation length are in good agreement with the predictions based on the field theory proposed by Janssen, Schmittmann, Leung, and Cardy. The theoretical predictions for the susceptibility and the magnetization are also well verified. We show that the transverse Binder parameter vanishes at the critical point in all dimensions d⩾2 and discuss how such result should be expected in the theory of Janssen et al. in spite of the existence of a dangerously irrelevant operator. Our results confirm the Gaussian nature of the transverse excitations. [ABSTRACT FROM AUTHOR]

We introduce a statistical ensemble for a single vortex filament of a three dimensional incompressible fluid. The core of the vortex is modeled by a quite generic stochastic process. We prove the existence of the partition function for both positive and a limited range of negative temperatures. [ABSTRACT FROM AUTHOR]