arXiv subjects

Massimiliano Gubinelli

Publications and source records attributed to Massimiliano Gubinelli.

At least 19 recordsLinked to original sources

arXiv2026

Refined global well-posedness for the periodic modulated Korteweg-de Vries equation

We revisit the pathwise global well-posedness issue of the modulated Korteweg-de Vries equation (KdV) on the circle. In the previous work (2024), by combining the $I$-method and the sewing lemma, the second and fourth authors with C. Chouk, G. Li, and J. Li proved its global well-posedness in negative Sobolev spaces. This result was, however, restricted to the scaling subcritical regime $s > - \frac 32$ due to the use of the classical KdV scaling. In this paper, by noting that the modulated KdV enjoys additional one degree of freedom in its scaling symmetry thanks to the modulation term, we apply a non-KdV scaling to the unknown and prove that, given any $s \in \mathbb R$, the modulated KdV on the circle with a sufficiently irregular modulation is globally well-posed in $H^s(\mathbb T)$, thus going beyond the barrier of the scaling critical regularity $s = - \frac 32$.

math.AP↗

arXiv2026

Nonlinear PDEs with modulated dispersion II: Korteweg-de Vries equation

(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.

math.AP↗

arXiv2026

Nonlinear PDEs with modulated dispersion III: multiplicative noises

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.

math.AP↗

arXiv2026

The FBSDE approach to sine-Gordon up to $6π$

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.

math-ph↗

arXiv2026

Nonlinear PDEs with modulated dispersion IV: normal form approach and unconditional uniqueness

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 $H^{\frac 16}(\mathbb T)$.

math.AP↗

arXiv2026

On the weak coupling limit of the periodic quantum Lorentz gas

We report partial progress on the weak coupling limit behavior of observables for the periodic quantum Lorentz gas. Our results indicate that for certain observables, the limit behavior is trivial and can be described via a transport equation, while for other observables, the existence of the limit hinges on the regularity properties at resonant momenta of a certain Bloch-Wigner transform. We are currently unable to prove or disprove this regularity property, and so the weak coupling limit for these observables remains an open question. A novelty of this work is the use of the sewing lemma in the derivation of the kinetic scaling limit for almost every mometum.

math-ph↗

arXiv2025

Large field problem in coercive singular PDEs

We derive a priori estimates for singular differential equations of the form \[ \mathcal{L} ϕ= P(ϕ,\nablaϕ) + f(ϕ,\nablaϕ)ξ\] where $P$ is a polynomial, $f$ is a sufficiently well-behaved function, and $ξ$ 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 $ξ=0$ 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 $ξ$ is small.

math.AP↗

arXiv2025

Parabolic stochastic quantisation of the fractional $Φ^4_3$ model in the full subcritical regime

We present a construction of the fractional $Φ^4$ 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. The measure is invariant under translations, reflection positive and has quartic exponential tails.

math.PR↗

arXiv2024

A simple construction of the sine-Gordon model via stochastic quantization

We present a simple PDE construction of the sine-Gordon measure below the first threshold ($\be^2 < 4π$), 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 $\be^2 < 2π$.

math.PR↗

arXiv2024

A stochastic analysis of subcritical Euclidean fermionic field theories

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.

math.PR↗

arXiv2024

Decay of correlations in stochastic quantization: the exponential Euclidean field in two dimensions

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 $\exp (αϕ)_2$) 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.

math-ph↗

arXiv2023

Gaussian Fluctuations for the stochastic Burgers equation in dimension $d\geq 2$

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.

math.PR↗

arXiv2023

Wilson-Itô diffusions

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.

math.PR↗

arXiv2023

Mixing for generic rough shear flows

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 $B^α_{1, \infty}$, $α\in (0, 1)$. We provide upper and lower bounds, showing that in general inviscid mixing in $H^{1/2}$ holds sharply with rate $r(t) \sim t^{1/(2 α)}$, while enhanced dissipation holds with rate $r(ν) \sim ν^{α/ (α+2)}$. Our results in the inviscid mixing case rely on the concept of $ρ$-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).

math.AP↗

arXiv2023

Non-commutative $L^{p}$ spaces and Grassmann stochastic analysis

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 $Φ^{4}_{2}$ 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 $Φ^{4}_{2}$.

math.PR↗

arXiv2023

Prevalence of $ρ$-irregularity and related properties

We show that generic Hölder continuous functions are $ρ$-irregular. The property of $ρ$-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 $ρ$-irregular and study in detail the geometric and analytic properties of $ρ$-irregular functions.

math.PR↗

arXiv2022

A singular integration by parts formula for the exponential Euclidean QFT on the plane

We give a novel characterization of the Euclidean quantum field theory with exponential interaction $ν$ 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.

math.PR↗

arXiv2022

On the variational method for Euclidean quantum fields in infinite volume

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 $φ^4_2$ 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 $P (φ)_2$ theories. We consider also the $\exp (βφ)_2$ model for $β^2 < 8 π$ (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 $P (φ)_2$ theories is lacking due to the difficulty of studying the stability of the wFBSDE against local perturbations.

math.PR↗