Person

Martin Hairer

ERGODICITY OF 2D SINGULAR STOCHASTIC NAVIER–STOKES EQUATIONS

We consider the 2D stochastic Navier–Stokes equations driven by noise that have the regularity of space-time white noise but don’t exactly coincide with it. We show that, provided that the intensity of the noise is sufficiently weak at high frequencies, th ...

2025

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

Martin Hairer

We present a simple PDE construction of the sine‐Gordon measure below the first threshold (), in both the finite and infinite volume settings, by studying the corresponding parabolic sine‐Gordon model. We also establish pathwise global well‐posedness of th ...

Wiley2025

A scaling limit of the 2D parabolic Anderson model with exclusion interaction

Martin Hairer

We consider the (discrete) parabolic Anderson model ∂u(t, x)/∂t = ∆u(t, x) + ξt(x)u(t, x), t ≥ 0, x ∈ Z d. Here, the ξ-field is R-valued, acting as a dynamic random environment, and ∆ represents the discrete Laplacian. We focus on the case where ξ is given ...

Institute of Mathematical Statistics2025

Periodic space-time homogenisation of the φ24 equation

Martin Hairer

We consider the homogenisation problem for the φ42 equation on the torus T2, namely the behaviour as ε →0 of the solutions to the equation suggestivelywritten as ∂tuε − ∇ · A(x/ε, t/ε2)∇uε = −u3ε + ξ where ξ deno ...

ACADEMIC PRESS INC ELSEVIER SCIENCE2025

Fluctuations of stochastic PDEs with long-range correlations

Martin Hairer

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions d≥3 with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-in ...

2025

SPECTRAL GAP FOR PROJECTIVE PROCESSES OF LINEAR SPDES

Martin Hairer

The goal of this work is to initiate the study of lower bounds for Lyapunov exponents of stochastic partial differential equations(SPDEs).To this end, we consider as a toy model the angular component πt=ut/|| ut|| associated to the solution u of a vector-v ...

2025

A Dynamical Yukawa2 Model

Martin Hairer

We prove local (in space and time) well-posedness for a mildly regularised version of the stochastic quantisation of the Yukawa(2) Euclidean field theory with a self-interacting boson. Our regularised dynamic is still singular but avoids non-local divergen ...

SPRINGER2025

Global existence for perturbations of the 2D stochastic Navier-Stokes equations with space-time white noise

Martin Hairer

We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations partial derivative( t)u + u center dot del u = Delta u - del p + sigma + xi, u(0, center dot ) = u(0),div (u) = 0, driven by additive space-time white noi ...

Springernature2024

Stochastic quantisation of Yang-Mills-Higgs in 3D

Martin Hairer

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space S is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the ...

SPRINGER HEIDELBERG2024

The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality

Martin Hairer

We provide a relatively compact proof of the BPHZ theorem for regularity structures of decorated trees in the case where the driving noise satisfies a suitable spectral gap property, as in the Gaussian case. This is inspired by the recent work (Linares et ...

Springer2024