arXiv subjects

Martin Hairer

Publications and source records attributed to Martin Hairer.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotics of Lyapunov Exponents and Phase Transitions for Fluids with Degenerate Forcing

In this paper we analyze the Lyapunov exponents of a slow-fast system where the slow component is an Ornstein-Uhlenbeck process which perturbs the linear evolution of a fast variable through a bilinear form. These naturally arise in many finite-dimensional models for turbulence such as Galerkin truncation of 2D Navier-Stokes, the Lorenz 96 system, and the Lorenz 63 system. Using our general results about the slow-fast system, we are able to prove phase transitions in the ergodicity of each of these models when degenerate stochastic forcing is applied: as a parameter (e.g. noise strength or viscosity) varies, the number of invariant measures of the system switches from one to several. We are also able to obtain precise asymptotics for the top Lyapunov exponent associated to the unstable invariant measure. The crux of our proof is using a Wiener chaos expansion to show that mass quickly transfers from stable modes to unstable ones.

math.PR↗

arXiv2026

Formation of clusters and coarsening in weakly interacting diffusions

This paper studies the clustering behavior of weakly interacting diffusions under the influence of sufficiently localized attractive interaction potentials on the one-dimensional torus. We describe how this clustering behavior is closely related to the presence of discontinuous phase transitions in the mean-field PDE. For local attractive interactions, we employ a new variant of the strict Riesz rearrangement inequality to prove that all global minimizers of the free energy are either uniform or single-cluster states, in the sense that they are symmetrically decreasing. We analyze different timescales for the particle system and the mean-field (McKean-Vlasov) PDE, arguing that while the particle system can exhibit coarsening by both coalescence and diffusive mass exchange between clusters, the clusters in the mean-field PDE are unable to move and coarsening occurs via the mass exchange of clusters. By introducing a new model for this mass exchange, we argue that the PDE exhibits dynamical metastability. We conclude by presenting careful numerical experiments that demonstrate the validity of our model.

math.AP↗

arXiv2026

How does the supercritical GMC converge?

In the spirit of [M. Biskup & O. Louidor, Adv. Math. 330 (2018)], we study the local structure of $\star$-scale invariant fields -- a class of log-correlated Gaussian fields -- around their extremal points by characterising the law of the "shape" of the field's configuration near such points. As a consequence, we obtain a refined understanding of the freezing phenomenon in supercritical Gaussian multiplicative chaos.

math.PR↗

arXiv2026

First Proof

To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.

cs.AI↗

arXiv2026

Singularity of solutions to singular SPDEs

Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $Φ^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $Φ^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.

math.PR↗

arXiv2025

Scaling Limits of a Weakly Perturbed Random Interface Model

We consider a random interface model on the discrete torus with $2n$ sites, obtained from the classical corner flip dynamics but with a weak global perturbation, namely an asymmetry of order $n^{-γ}$ of the direction of growth that switches direction based on the sign of the total area under the interface. The slopes of this model can be viewed as a non-simple exclusion process at half filling with globally dependent rates. We show that, for $γ=1$, the hydrodynamic equation of the empirical density is given by a time concatenation of the viscous Burgers equation and the heat equation. Moreover, for $n$ prime and $γ>\frac{6}{7}$, we establish convergence in law of the equilibrium fluctuations to an infinite-dimensional Ornstein-Uhlenbeck process.

math.PR↗

arXiv2025

Uniqueness of supercritical Gaussian multiplicative chaos

We show that, for general convolution approximations to a large class of log-correlated Gaussian fields, the properly normalised supercritical Gaussian multiplicative chaos measures converge stably to a nontrivial limit. This limit depends on the choice of regularisation only through a multiplicative constant and can be characterised as an integrated atomic measure with a random intensity expressed in terms of the critical Gaussian multiplicative chaos.

math.PR↗

arXiv2025

Homogenisation of singular SPDEs

We introduce an approach to study homogenisation of a large class of singular SPDEs of the form $$ \partial_t u_\varepsilon - \nabla\cdot {A}(x/\varepsilon,t/\varepsilon^2) \nabla u_\varepsilon = F(x/\varepsilon , t/\varepsilon^2, u_\varepsilon , \nabla u_\varepsilon , ξ) $$ which is based on the idea of importing (classical) homogenisation results into the framework of regularity structures and the insight that one can rewrite the SPDE under consideration in terms of a model, where the correctors (from homogenisation theory) are seen as further `abstract noises'. As applications, we establish periodic space-time homogenisation results for oscillatory generalisations of the 2d g-PAM and $Φ^4_3$ equation proving that when the noise is regularised at scale $δ\ll 1$ solutions to the equation with coefficient field ${A}(x/\varepsilon,t/\varepsilon^2)$, when appropriately renormalised, converge to solutions to the corresponding homogenised equation along any sequence $(\varepsilon,δ)\to 0$. We make the observation that the unbounded divergences can be written as sums of two types of terms: `small scale' terms, the spatial dependence of which is an explicit local function of the coefficient field and `large scale' terms, which for logarithmic divergences are explicit involving the homogenised matrix and correctors. Furthermore, in order to recover the same solution to the corresponding homogenised equation along any joint limit $(\varepsilon, δ)\to 0$ one has to subtract additional bounded renormalisation constants which appear due to oscillations at mesoscopic scales, as well as due to resonances between the coefficient field and the oscillations in the nonlinearity.

math.AP↗

arXiv2025

Quasi-Gaussianity of the 2D stochastic Navier-Stokes equations

We study the qualitative properties of solutions to the 2D stochastic Navier-Stokes equations with forcing that is white in time and coloured in space. Our main result shows that the unique invariant measure of this system is equivalent to that of the corresponding Ornstein-Uhlenbeck process. Our method relies on a generalization of the "time-shifted Girsanov method" of [MS05, MRS22] to compare the laws of time marginals for dissipative SPDEs. This generalisation allows to not only compare solutions to a nonlinear equation to those of the corresponding linear equation, but also to directly compare two nonlinear equations. We use this to establish equivalence of the Navier-Stokes system to a "twisted" nonlinear system that leaves the Gaussian measure invariant. We further apply this method to establish similar equivalence statements for a family of hypoviscous Navier-Stokes equations.

math.PR↗

arXiv2025

A critical stochastic heat equation with long-range noise

We consider a semilinear stochastic heat equation in spatial dimension at least $3$, forced by a noise that is white in time with a covariance kernel that decays like $\lvert x\rvert^{-2}$ as $\lvert x\rvert\to\infty$. We show that in an appropriate diffusive scaling limit with a logarithmic attenuation of the noise, the pointwise statistics of the solution can be approximated by the solution to a forward-backward stochastic differential equation (FBSDE). The scaling and structure of the problem is similar to that of the two-dimensional stochastic heat equation forced by an approximation of space-time white noise considered by the first author and Gu (Ann. Probab., 2022). However the resulting FBSDE is different due to the long-range correlations of the noise.

math.PR↗

arXiv2025

Noncommutative Regularity Structures

We extend the theory of regularity structures [Hai14] to allow processes belonging to locally $m$-convex topological algebras. This extension includes processes in the locally $C^{*}$-algebras of [CHP25] used to localise singular stochastic partial differential equations involving fermions, as well as processes in Banach algebras such as infinite-dimensional semicircular\circular Brownian motion, and more generally the $q$-Gaussians of [BS91, BKS97, Boż99]. A new challenge we encounter in the $q$-Gaussian setting with $q \in (-1,1)$ are noncommutative renormalisation estimates where we must estimate operators in homogeneous $q$-Gaussian chaoses with arbitrary operator insertions. We introduce a new Banach algebra norm on $q$-Gaussian operators that allows us to control such insertions; we believe this construction could be of independent interest.

math.PR↗

arXiv2025

Ergodicity of 2D singular stochastic Navier-Stokes equations

We consider the 2D stochastic Navier-Stokes equations driven by noise that has the regularity of space-time white noise but doesn't exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at high frequencies, this systems admits uniform bounds in time, so that it has an invariant measure, for which we obtain stretched exponential tail bounds.

math.PR↗

arXiv2025

Ergodicity of infinite volume $Φ^4_3$ at high temperature

We consider the infinite volume $Φ^4_3$ dynamic and show that it is globally well-posed in a suitable weighted Besov space of distributions. At high temperatures / small coupling, we furthermore show that the difference between any two solutions driven by the same realisation of the noise converges to zero exponentially fast. This allows us to characterise the infinite-volume $Φ^4_3$ measure at high temperature as the unique invariant measure of the dynamic, and to prove that it satisfies all Osterwalder--Schrader axioms, including invariance under translations, rotations, and reflections, as well as exponential decay of correlations.

math.PR↗

arXiv2025

A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise

We prove a stochastic version of the classical RAGE theorem that applies to the two-point motion generated by noisy transport equations. As a consequence, we identify a necessary and sufficient condition for the corresponding diffusive equation to be dissipation enhancing. This involves the identification of a non-trivial, finite dimensional subspace that is invariant for the family of self-adjoint operator characterizing the structure of the transport noise. We discuss several examples and prove a sharp enhanced dissipation rate for stochastic shear flows.

math.AP↗

arXiv2025

A noise-induced transition in the Lorenz system

We consider a stochastic perturbation of the classical Lorenz system in the range of parameters for which the origin is the global attractor. We show that adding noise in the last component causes a transition from a unique to exactly two ergodic invariant measures. The bifurcation threshold depends on the strength of the noise: if the noise is weak, the only invariant measure is Gaussian, while strong enough noise causes the appearance of a second ergodic invariant measure.

math.PR↗

arXiv2025

Probabilistic interpretation of quantum field theories

In this note we provide a gentle introduction to the concepts and intuition behind the recent breakthrough results on the mathematically rigorous construction of a non-trivial 2D conformal field theory, namely the so-called Liouville theory. This gives us the opportunity to review Segal's axioms for conformal field theories and to discuss in some detail how the free field fits into them.

hep-th↗

arXiv2025

Fluctuations of stochastic PDEs with long-range correlations

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of $|x|^{-κ}$ at infinity, where $κ\in (2, d)$. Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree $-κ$. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies.

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↗