Papers

We present a new, simple and self-contained technique for proving a mass gap for Euclidean quantum field theories in the high-temperature regime using the stochastic quantisation approach. As an application, we establish exponential decay of spatial correlations for the fractional \(\Phi^{4}\) model on the plane in the Wick-ordering regime. The key idea is to analyse the propagation of perturbations using the linearised stochastic quantisation equation. Spatial propagation is controlled by an auxiliary random walk, while the potentially destabilising effect of the renormalised nonlinearity is controlled by combining dissipative energy estimates for the nonlinear dynamic with the Boué–Dupuis variational formula.

We develop a rigorous framework for the description of general local composite observables in Euclidean quantum field theories. The approach is nonperturbative and combines renormalisation group ideas with techniques of stochastic analysis. Via renormalisation dictated by sufficiently precise approximate solution theory of corresponding flow equations, we construct various local observables of the sine–Gordon model on the full space up to \(\beta^{2}6\pi\), including highly singular ones such as vertex fields \(exp(i\alpha\phi(y))\) in the optimal Wick regime \(|\Re e(\alpha)|4\pi/\beta\) and derivatives of arbitrarily high order, i.e. \(\partial^{\mu}\phi(y)\) for any multiindex \(\mu\in\mathbb{N}^{2}_{0}\). Applications of the approach include a direct proof of smoothness of observable-inserted correlation functions, and verification that the constructed objects satisfy a version of the prefactorization algebra axioms adapted to our setting.

We investigate pathwise well-posedness of the stochastic modulated Korteweg-de Vries equation (KdV) on the circle with a multiplicative noise, where a time non-homogeneous modulation acts on the linear dispersion term. (i) In the Young case (= fractional-in-time case with Hurst parameter greater than \(\frac 12\)), we establish a new regularization-by-noise phenomenon on the stochastic convolution in a pathwise manner, where a gain of spatial regularity becomes (arbitrarily) larger for more irregular modulations. We then prove that, given any \(s \in \mathbb R\) and any multiplicative Young noise, however rough it is in space, the stochastic modulated KdV is pathwise locally well-posed in \(H^s(\mathbb T)\), provided that the modulation is sufficiently irregular. (ii) In the rough case (= white-in-time case), irregularity of the modulation does not induce any smoothing on the stochastic convolution, and in fact, there is a slight loss in the spatial regularity. In this case, by slightly regularizing the multiplicative noise term, we prove pathwise local well-posedness in \(H^s(\mathbb T)\) for any given \(s \in \mathbb R\), provided that the noise is sufficiently smooth in space. We achieve these goals by combining (i) the sewing lemma approach to the nonlinear Young integration theory, introduced by Chouk and the second author (2014), and (ii) the pathwise construction of stochastic convolutions as Young or rough integrals via the random tensor estimate and the sewing lemma, introduced by the first, fourth, and fifth authors (2026). In the appendix, we also present an example of regularization by noise for a stochastic modulated Schrödinger equation with a multiplicative Young noise.

We derive a priori estimates for singular differential equations of the form \(\mathcal{L} \phi = P(\phi,\nabla\phi) + f(\phi,\nabla\phi)\xi\) where \(P\) is a polynomial, \(f\) is a sufficiently well-behaved function, and \(\xi\) is an irregular distribution such that the equation is subcritical. The differential operator \(\mathcal L\) is either a derivative in time, in which case we interpret the equation using rough path theory, or a heat operator, in which case we interpret the equation using regularity structures. Our only assumption on \(P\) is that solutions with \(\xi=0\) exhibit coercivity. Our estimates are local in space and time, and independent of boundary conditions. One of our main results is an abstract estimate that allows one to pass from a local coercivity property to a global one using scaling, for a large class of equations. This allows us to reduce the problem of deriving a priori estimates to the case when \(\xi\) is small.

We study the modulated Korteweg-de~Vries equation (KdV) on the circle with a time non-homogeneous modulation acting on the linear dispersion term. By adapting the normal form approach to the modulated setting, we prove sharp unconditional uniqueness of solutions to the modulated KdV in \(L^2(\mathbb T)\) if a modulation is sufficiently irregular. For example, this result implies that if the modulation is given by a sample path of a fractional Brownian motion with Hurst index \(0 H \frac 25\), the modulated KdV on the circle is unconditionally well-posed in \(L^2(\mathbb T)\). Our normal form approach provides the construction of solutions to the modulated KdV (and the associated nonlinear Young integral) {\it without} assuming any positive regularity in time. As an interesting byproduct of our normal form approach, we extend the construction of the nonlinear Young integral to a much larger class of functions, and obtain an improved Euler approximation scheme as compared to the classical sewing lemma approach. We also establish analogous sharp unconditional uniqueness results for the modulated Benjamin-Ono equation and the modulated derivative nonlinear Schrödinger equation (NLS) with a quadratic nonlinearity. In the appendix, we prove sharp unconditional uniqueness of the cubic modulated NLS on the circle in \(H^{\frac 16}(\mathbb T)\).

We present a simple PDE construction of the sine-Gordon measure below the first threshold (\({\beta}^2 4\pi\)), in both the finite and infinite volume settings, by studying the corresponding parabolic sine-Gordon model. We also establish pathwise global well-posedness of the hyperbolic sine-Gordon model in finite volume for \({\beta}^2 2\pi\).

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.

We introduce Wilson-Itô 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 à 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.

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 Høegh–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.

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 Itô 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 \(\Phi^{4}_{2}\).

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.

We present a construction of the measure 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 a truncated flow equation for the effective description of the model at sufficiently small scales and on coercive estimates for the nonlinear stochastic partial differential equation describing the interacting field. The constructed measure is translation invariant, reflection positive, and has quartic exponential tails.

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.

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.

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.

We study mixing and diffusion properties of passive scalars driven by \(generic\) rough shear flows. Genericity is here understood in the sense of prevalence and (ir)regularity is measured in the Besov-Nikolskii scale \(B^{\alpha}_{1, \infty}\), \(\alpha \in (0, 1)\). We provide upper and lower bounds, showing that in general inviscid mixing in \(H^{1/2}\) holds sharply with rate \(r(t) \sim t^{1/(2 \alpha)}\), while enhanced dissipation holds with rate \(r(\nu) \sim \nu^{\alpha / (\alpha+2)}\). Our results in the inviscid mixing case rely on the concept of \(\rho\)-irregularity, first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and provide some new insights compared to the behavior predicted by Colombo, Coti Zelati and Widmayer (arXiv:2009.12268, 2020).

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 \(C^{\ast}\)-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.

We construct the \(\Phi^4_3\) measure on a periodic three dimensional box as an absolutely continuous perturbation of a random shift of the Gaussian free field. The shifted measure is constructed via Girsanov's theorem and the relevant filtration is the one generated by a scale parameter. As a byproduct we give a self-contained proof that the \(\Phi^4_3\) measure is singular wrt. the Gaussian free field.

We show that generic Hölder continuous functions are \(\rho\)-irregular. The property of \(\rho\)-irregularity has been first introduced by Catellier and Gubinelli (Stoc. Proc. Appl. 126, 2016) and plays a key role in the study of well-posedness for some classes of perturbed ODEs and PDEs. Genericity here is understood in the sense of prevalence. As a consequence we obtain several results on regularisation by noise "without probability", i.e. without committing to specific assumptions on the statistical properties of the perturbations. We also establish useful criteria for stochastic processes to be \(\rho\)-irregular and study in detail the geometric and analytic properties of \(\rho\)-irregular functions.

We analyse the effect of a generic continuous additive perturbation to the well-posedness of ordinary differential equations. Genericity here is understood in the sense of prevalence. This allows us to discuss these problems in a setting where we do not have to commit ourselves to any restrictive assumption on the statistical properties of the perturbation. The main result is that a generic continuous perturbation renders the Cauchy problem well-posed for arbitrarily irregular vector fields. Therefore we establish regularisation by noise "without probability".

We prove existence and uniqueness of martingale solutions to a (slightly) hyperviscous stochastic Navier-Stokes equation in 2d with initial conditions absolutely continuous with respect to the Gibbs measure associated to the energy, getting the results both in the torus and in the whole space setting.

We obtain a dimensional reduction result for the law of a class of stochastic differential equations using a supersymmetric representation first introduced by Parisi and Sourlas.

We study a class of elliptic SPDEs with additive Gaussian noise on \(\mathbb{R}^2 \times M\), with \(M\) a \(d\)-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential \(V\), convex at infinity and growing at most exponentially. For quite general coefficients and a suitable regularity of the noise we obtain, via the dimensional reduction principle discussed in our previous paper on the topic, the identity between the law of the solution to the SPDE evaluated at the origin with a Gibbs type measure on the abstract Wiener space \(L^2 (M)\). The results are then applied to the elliptic stochastic quantization equation for the scalar field with polynomial interaction over \(\mathbb{T}^2\), and with exponential interaction over \(\mathbb{R}^2\) (known also as Høeg-Krohn or Liouville model in the literature). In particular for the exponential interaction case, the existence and uniqueness properties of solutions to the elliptic equation over \(\mathbb{R}^{2 + 2}\) is derived as well as the dimensional reduction for the values of the “charge parameter” \(\sigma = \frac{\alpha}{2\sqrt{\pi}} \sqrt{4 \left( 8 - 4 \sqrt{3} \right) \pi} \simeq \sqrt{4.23\pi}\), for which the model has an Euclidean invariant probability measure (hence also permitting to get the corresponding relativistic invariant model on the two dimensional Minkowski space).

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in \(\mathbb{R}^2\). This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in \(d + 2\) dimension with a Gibbs measure in \(d\) dimensions. This phenomenon is similar to the relation between an \(\mathbb{R}^{d + 1}\) dimensional parabolic SPDE and its \(\mathbb{R}^d\) dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our \(d = 0\) context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the Gaussian case.

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.

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 [GP18a] to a wider class of equations.

We present a new construction of the Euclidean \(\Phi^4\) quantum field theory on \(\mathbb{R}^3\) based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on \(\mathbb{R}^3\) defined on a periodic lattice of mesh size \(\varepsilon\) and side length \(M\). We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as \(\varepsilon \rightarrow 0\), \(M \rightarrow \infty\). 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 \emphdistribution on the space of Euclidean fields.

We analyze nonlinear Schrödinger and wave equations whose linear part is given by the renormalized Anderson Hamiltonian in two and three dimensional periodic domains.

We introduce an explicit description of the \(\Phi^4_3\) measure on a bounded domain. Our starting point is the interpretation of its Laplace transform as the value function of a stochastic optimal control problem along the flow of a scale regularization parameter. Once small scale singularities have been renormalized by the standard counterterms, \(\Gamma\)-convergence allows to extend the variational characterization to the unregularized model.

We prove existence of global solutions to singular SPDEs on \(\mathbb{R}^d\) 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 \(\Phi^4_d\) Euclidean quantum field theory via Parisi–Wu stochastic quantization, while the elliptic equations are linked to the \(\Phi^4_{d-2}\) Euclidean quantum field theory via the Parisi–Sourlas dimensional reduction mechanism.

We establish the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on a three dimensional periodic domain to the dynamic \(\Phi^4_3\) model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular \(C^9\) is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their \(L^p\) norms in terms of the graphs of the standard \(\Phi^4_3\) stochastic terms.

We study the two-dimensional stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing. In particular, we introduce a time-dependent renor- malization and prove that SNLW is pathwise locally well-posed. As an application of the local well-posedness argument, we also establish a weak universality result for the renormalized SNLW.

We review recent results on the analysis of singular stochastic partial differential equations in the language of paracontrolled distributions.

We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of [Gubinelli, Perkowski 2016], but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of [Gubinelli, Perkowski 2016] to a non-stationary regime.

We introduce a non-linear paracontrolled calculus and use it to renormalise a class of singular SPDEs including certain quasilinear variants of the periodic two dimensional parabolic Anderson model.

We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some ideas we introduced in a previous work dealing with rough kinetic PDEs [arXiv:1604.00437].

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 Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.

We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman–Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well–posedness for the corresponding kinetic formulation.

We use the notion of energy solutions of the stochastic Burgers equation to give a short proof of the Hairer-Quastel universality result for a class of stationary weakly asymmetric stochastic PDEs.

The Kardar-Parisi-Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the KPZ equation on the real line by showing that its energy solutions (as introduced by Gonçalves and Jara and later refined by Gubinelli and Jara) are unique. Together with various convergence results already present in the literature, this establishes the weak KPZ universality conjecture for a wide class of models. A remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole-Hopf solution, but it involves an additional drift \(t/12\).

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 KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Goncalves and Jara and paracontrolled solutions. 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 which does not rely on the Cole-Hopf transform.

These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non–linear PDEs with stochastic and irregular terms. We explain how it is possible to handle them using two main techniques: the notion of energy solutions and that of paracontrolled distributions. In order to maintain a link with physical intuitions, we motivate such singular SPDEs via a homogenisation result for a diffusion in a random potential.

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.

(Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here. See the abstract in the paper.) We study dispersive equations with a time non-homogeneous modulation acting on the linear dispersion term. As primary models, we consider the Korteweg-de Vries equation (KdV) and related equations such as the Benjamin-Ono equation (BO) and the intermediate long wave equation (ILW), imposing certain irregularity conditions on the time non-homogeneous modulation. In this work, we establish phenomena called regularization by noise in three-folds: (i) When the modulation is sufficiently irregular, we show that the modulated KdV on both the circle and the real line is locally well-posed in the regime where the (unmodulated) KdV equation is known to be ill-posed. In particular, given any \(s \in \mathbb R\), we show that the modulated KdV on the circle with a sufficiently irregular modulation is locally well-posed in \(H^s(\mathbb T)\). Moreover, by adapting the \(I\)-method to the current modulated setting, we prove global well-posedness of the modulated KdV in negative Sobolev spaces. (ii) It is known that certain (semilinear) dispersive equations such as BO and ILW exhibit quasilinear nature. We show that sufficiently irregular modulations make the modulated versions of these equations semilinear by establishing their local well-posedness by a contraction argument, providing local Lipschitz continuity of the solution map. (iii) We also prove nonlinear smoothing for these modulated equations, where we show that a gain of regularity of the nonlinear part becomes (arbitrarily) larger for more irregular modulations. As applications of our approach, we also include further examples.

Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main application is to give a path by path construction of the stochastic integral in the plane and obtain a stratonovich change of variables formula.

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.

In this paper linear stochastic transport and continuity equations with drift in critical \(L^{p}\) spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their \(C^\alpha\) regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

Starting from the N-particle Nelson Hamiltonian defined by imposing an ultraviolet cutoff, we perform ultraviolet renormalization by showing that in the zero cutoff limit a self-adjoint operator exists after a logarithmically divergent term is subtracted from the original Hamiltonian. We obtain this term as the diagonal part of a pair interaction appearing in the density of a Gibbs measure derived from the Feynman-Kac representation of the Hamiltonian. Also, we show existence of a weak coupling limit of the renormalized Hamiltonian and derive an effective Yukawa interaction potential between the particles.

We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of "irregularity" for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well–posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularising effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.

We consider a stochastic linear transport equation with a globally Hölder continuous and bounded vector field. Opposite to what happens in the deterministic case where shocks may appear, we show that the unique solution starting with a C^1-initial condition remains of class \(C^{1}\) in space. We also improve some results of Flandoli-Gubinelli-Priola (2009) about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.

We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.

We study a generalized 1d periodic SPDE of Burgers type: \( \partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi \) where \(\theta 1/2\), \(-A\) is the 1d Laplacian, \(\xi\) is a space-time white noise and the initial condition \(u_0\) is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when \(\theta1/2\). When \(\theta5/4\) we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of the same equation and for a model of stationary 2d stochastic Navier-Stokes evolution.

We consider the ordinary differential equation (ODE) \(dx_{t} =b(t,x_{t} ) dt+ dw_{t}\) where \(w\) is a continuous driving function and \(b\) is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path \(w\) on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of \(\rho\)-\tmtextitirregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function \(w\) sampled according to the law of the fractional Brownian motion of Hurst index \(H \in (0,1)\), we prove that almost surely the ODE admits a solution for all \(b\) in the Besov-HÃ{\P}lder space \(B^{\alpha+1}_{\infty , \infty}\) with \(\alpha -1/2H\). If \(\alpha 1-1/2H\) then the solution is unique among a natural set of continuous solutions. If \(H1/3\) and \(\alpha 3/2-1/2H\) or if \(\alpha 2-1/2H\) then the equation admits a unique Lipschitz flow. Note that when \(\alpha 0\) the vector field \(b\) is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.

We prove the existence of a global solution for the filament equation with inital condition given by a geometric rough path in the sense of Lyons (1998).Our work gives a positive answer to a question left open in recent publications: Berselli and Gubinelli (2007) showed the existence of global solution for a smooth initial condition while Bessaih, Gubinelli, Russo (2005) proved the existence of a local solution for a general initial condition given by a rough path.

The motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears.

This article is devoted to define and solve an evolution equation of the form \(dy_t=\Delta y_t dt+ dX_t(y_t)\), where \(\Delta\) stands for the Laplace operator on a space of the form \(L^p(\mathbb{R}^n)\), and \(X\) is a finite dimensional noisy nonlinearity whose typical form is given by \(X_t(\varphi)=\sum_{i=1}^N x^{i}_t f_i(\varphi)\), where each \(x=(x^{(1)},…,x^{(N)})\) is a \(\gamma\)-Hölder function generating a rough path and each \(f_i\) is a smooth enough function defined on \(L^p(\mathbb{R}^n)\). The generalization of the usual rough path theory allowing to cope with such kind of systems is carefully constructed.

We consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Holder continuous drift term. We prove existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Ito-Tanaka type. The proof requires non-standard elliptic estimates in Holder spaces. As an application of the stochastic flow, we obtain a Bismut-Elworthy-Li type formula for the first derivatives of the associated diffusion semigroup.

We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential equation with bounded coefficients.

This is a review paper on recent work about the connections between rough path theory, the Connes-Kreimer Hopf algebra on rooted trees and the analysis of finite and infinite dimensional differential equation. We try to explain and motivate the theory of rough paths introduced by T. Lyons in the context of differential equations in presence of irregular noises. We show how it is used in an abstract algebraic approach to the definition of integrals over paths which involves a cochain complex of finite increments. In the context of such abstract integration theories we outline a connection with the combinatorics of rooted trees. As interesting examples where these ideas apply we present two infinite dimensional dynamical systems: the Navier-Stokes equation and the Korteweg-de-Vries equation.

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a Brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the Brownian character of the model on large scale and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties.

We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space–time Gaussian noise with singular space covariance and Brownian time dependence.

Motivated by applications to quantum field theory we consider Gibbs measures for which the reference measure is Wiener measure and the interaction is given by a double stochastic integral and a pinning external potential. In order properly to characterize these measures through DLR equations, we are led to lift Wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields. We prove existence and basic properties of such Gibbs measures in the small coupling regime by means of cluster expansion.

The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg–de Vries (KdV) equation on a periodic domain and with initial condition in \({\mathcal{F}} L^{\alpha,p}\) spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.

In this note we study a class of specifications over \(d\)-dimensional Wiener measure which are invariant under uniform translation of the paths. This degeneracy is removed by restricting the measure to the \(\sigma\)-algebra generated by the increments of the coordinate process. We address the problem of existence and uniqueness of Gibbs measures and prove a central limit theorem for the rescaled increments. These results apply to the study of the ground state of the Nelson model of a quantum particle interacting with a scalar boson field.

We establish a representation of a class of solutions of 3d Navier-Stokes equations in \({\mathbb R}^3\) using sums over rooted trees. We study the convergence properties of this series recovering in a simplified manner some results obtained recently by Sinai and other known results for solutions in spaces of pseudo-measures introduced initially by Le Jan and Sznitman. The series representation make sense also in the critical case where there exists global solutions for small initial data and it allows the study of their long-time or small-distance behavior.

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 interpret our results as a basis to theoretical validation of numerical methods used in state-of-the-art computations of turbulent flows.

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: - Various numerical observations that suggest that the critical line lies strictly in between the two bounds. - A rigorous statistical test based on concentration inequalities and super-additivity, for determining whether a given point of the phase diagram is in the localized phase. This is applied in particular to show that, with a very low level of error, the lower bound does not coincide with the critical line. - An analysis of the precise asymptotic behavior of the partition function in the delocalized phase, with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a Brownian scaling. - A new proof of the lower bound on the critical line. This proof relies on a characterization of the localized regime which is more appealing for interpreting the numerical data.

We investigate the critical behavior of the two-dimensional randomly driven lattice gas, in which particles are driven along one of the lattice axes by an infinite external field with randomly changing sign. A finite-size scaling (FSS) analysis provides novel evidences that this model is not in the same universality class as the driven lattice gas with a constant drive (DLG), contrarily to what has been recently reported in the literature. Indeed, the FSS functions of transverse observables (i.e., related to order-parameter fluctuations with wave vector perpendicular to the direction of the field) differ from the mean-field behavior predicted and observed within the DLG universality class. At variance with the DLG case, FSS is attained on lattices with fixed aspect ratio and anisotropy exponent equal to 1 and the transverse Binder cumulant does not vanish at the critical point.

A definition of K41 scaling law for suitable families of measures is given and investigated. First, a number of necessary conditions are proved. They imply the absence of scaling laws for 2D stochastic Navier-Stokes equations and for the stochastic Stokes (linear) problem in any dimension, while they imply a lower bound on the mean vortex stretching in 3D. Second, for 3D stochastic Navier-Stokes equations necessary and sufficient conditions for K41 are proved, translating the problem into bounds for energy and enstrophy of high and low modes respectively. The validity of such conditions in 3D remains open. Finally, a stochastic vortex model with such properties is presented.

In this note, we study the non-linear evolution problem \(dY_t = -A Y_t dt + B(Y_t) dX_t\), 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 \(\gamma1/2\)). 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.

We study an evolution problem in the space of continuous loops in three-dimensional Euclidean space modelled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting from Hölder regular loops with index greater than 1/3. When the Hölder regularity of the initial condition X is smaller or equal 1/2 we require X to be a rough path in the sense of Lyons. The solution will then live in an appropriate space of rough paths. In particular we can construct (local) solution starting from almost every Brownian loop.

A random field composed by Poisson distributed Brownian vortex filaments is constructed. The filament have a random thickness, length and intensity, governed by a measure \(\gamma\). Under appropriate assumptions on \(\gamma\) we compute the scaling law of the structure function and get the multifractal scaling as a particular case.

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 \(d\ge 2\) 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.

We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to irregular paths with Hoelder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons' theory of rough paths in Hoelder topology.

In a recent Letter Albano and Saracco study the dynamic critical behavior of some anisotropic driven lattice gases by Monte Carlo (MC) simulations. In this Comment we point out that the Ansätze they use to relate the measured scaling exponents with the critical exponents analytically computed within different field-theoretical approaches do not take properly into account the strongly anisotropic nature of the phase transition, by implicitly assuming \(z = z_{\bot} = z_{\parallel}\). As a consequence, at variance with the claims by the authors, their MC data are not conclusive to determine which one of the field theories proposed in the literature correctly describes the universal properties of the phase transition in these lattice gases.

We discuss the shape dependence of the finite-size scaling limit in a strongly anisotropic O(N) model in the large-N limit. We show that scaling is observed even if an incorrect value for the anisotropy exponent is considered. However, the related exponents may only be effective ones, differing from the correct critical exponents of the model. We discuss the implications of our results for numerical finite-size scaling studies of strongly anisotropic systems.

Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng structure. As to question b), it is shown that Hölder continuous paths with exponent \(\gamma 1/2\) define integral flat chains.

We define a transverse correlation length suitable to discuss the finite-size-scaling behavior of an out-of-equilibrium lattice gas, whose correlation functions decay algebraically with the distance. By numerical simulations we verify that this definition has a good infinite-volume limit independent of the lattice geometry. We study the transverse fluctuations as they can select the correct field-theoretical description. By means of a careful finite-size scaling analysis, without tunable parameters, we show that they are Gaussian, in agreement with the predictions of the model proposed by Janssen, Schmittmann, Leung, and Cardy.

We present a detailed 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, study its finite-size-scaling behavior and extrapolate it to the infinite-volume limit by using an iterative extrapolation method. The same method is applied to the susceptibility. We determine the transverse exponents \(\gamma\) and \(\nu\). They turn out to be in perfect agreement with the theoretical predictions. We also compute the transverse Binder parameter, which apparently vanishes at the critical point, and the distribution of the magnetization. These results confirm the Gaussian nature of the transverse excitations.

We address the problem of the definition of the finite-volume correlation length. First, we study the large-N limit of the N-vector model, and we show the existence of several constraints on the definition if regularity of the finite-size scaling functions and correct anomalous behaviour above the upper critical dimension are required. Then, we study in detail a model in which the zero mode in prohibited. Such a model is a generalization of the fixed-magnetization Ising model which is equivalent to the lattice gas. Also in this case, we find that the finite-volume correlation length must satisfy appropriate constraints in order to obtain regular finite-size scaling functions, and, above the upper critical dimension, an anomalous scaling behaviour. The large-N results are confirmed by a one-loop calculation in the lattice \(\phi^4\) theory.

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