arXiv subjects

Tom Klose

Publications and source records attributed to Tom Klose.

9 recordsLinked to original sources

arXiv2026

A PDE approach to the 2D Yang-Mills measure

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. One of our main results is a rough version of Uhlenbeck compactness for distributional connections based on rough additive functions. The main ingredient is a singular elliptic PDE to obtain a Coulomb gauge. We solve this PDE using regularity structures and a method inspired by the implicit function theorem. Surprisingly, the model needed to solve the equation is determined entirely from rough additive functions, which is a simpler and geometrically more natural object. We apply our results to the Yang-Mills measure on the unit square, showing that it has a gauge-fixed representation with optimal regularity. Although we focus on the unit square in this article, we expect our results to apply to more general surfaces.

math.AP↗

arXiv2026

A Rough Functional Breuer-Major Theorem

We extend the functional Breuer-Major theorem by Nourdin and Nualart (2020) to the space of rough paths. The proof of tightness combines the multiplication formula for iterated Malliavin divergences, due to Furlan and Gubinelli (2019), with Meyer's inequality and a Kolmogorov-type criterion for the r-variation of cadlag rough paths, due to Chevyrev et al. (2022). Since martingale techniques do not apply, we obtain the convergence of the finite-dimensional distributions through a bespoke version of Slutsky's lemma: First, we overcome the lack of hypercontractivity by an iterated integration-by-parts scheme which reduces the remaining analysis to finite Wiener chaos; crucially, this argument relies on Malliavin differentiability of the nonlinearity but not on chaos decay and, as a consequence, encompasses the centred absolute value function. Second, in the spirit of the law of large numbers, we show that the diagonal of the second-order process converges to an explicit symmetric correction term. Finally, we compute all the moments of the remaining process and, through a fine combinatorial analysis, show that they converge to those of the Stratonovich Brownian rough path perturbed by an antisymmetric area correction, as computed by a suitable amendment of Fawcett's theorem. All of these steps benefit from a major combinatorial reduction that is implied by the original argument of Breuer and Major (1983).

math.PR↗

arXiv2026

Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension

We study the large-scale behaviour of a class of driven diffusive systems modelled by a Stochastic Partial Differential Equation, the Stochastic Burgers Equation (SBE) with general nonlinearity, at the critical dimension and in infinite volume. Our main result shows that, under a logarithmically superdiffusive space-time scaling, it is given by the same explicit Gaussian Fixed point obtained in [G. Cannizzaro, Q. Moulard, & F. Toninelli, arxiv.org/abs/2501.00344, 2025] for the quadratic SBE, but with suitably renormalised coefficients, thereby rigorously justifying and partly correcting the classical Physics derivation of the SBE in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] based on Spohn's theory of nonlinear fluctuating hydrodynamics. Besides, ours is the first universality-type result for out-of-equilibrium systems and the first extension of [M. Hairer, J. Quastel, Forum of Mathematics, Pi, Vol. 6, 2018, e3], to the critical dimension and beyond weak coupling. The major challenge in our work is the mild growth condition on the nonlinearity which renders even the well-posedness of the microscopic equation non-trivial. Additional key novelties include the derivation of fine estimates on the non-quadratic part of the generator as well as a new approximation for the resolvent associated to the solution of the quadratic SBE.

math.PR↗

arXiv2025

Perturbative renormalisation of the $\Phi^4_{4-\varepsilon}$ model via generalized Wick maps

We consider the perturbative renormalisation of the $\Phi^4_d$ model from Euclidean Quantum Field Theory for any, possibly non-integer dimension $d<4$. The so-called BPHZ renormalisation, named after Bogoliubov, Parasiuk, Hepp and Zimmermann, is usually encoded into extraction-contraction operations on Feynman diagrams, which have a complicated combinatorics. We show that the same procedure can be encoded in the much simpler algebra of polynomials in two unknowns $X$ and $Y$, which represent the fourth and second Wick power of the field. In this setting, renormalisation takes the form of a \lq\lq Wick map\rq\rq\ which maps monomials into Bell polynomials. The construction makes use of recent results by Bruned and Hou on multiindices, which are algebraic objects of intermediate complexity between Feynman diagrams and polynomials.

math.PR↗

arXiv2024

Large Deviations of the $\Phi^4_3$ Measure via Stochastic Quantisation

The $\Phi^4_3$ measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970's has been one of the celebrated achievements of constructive quantum field theory. In recent years, progress in the field of singular stochastic PDEs, initiated by the theory of regularity structures, has allowed for a new construction of the $\Phi^4_3$ EQFT as the invariant measure of a previously ill-posed Langevin dynamics, a strategy originally proposed by Parisi and Wu ('81) under the name stochastic quantisation. We apply the same methodology to obtain a large deviation principle (LDP) for the family of periodic $\Phi^4_3$ measures at varying temperature. In addition, we show that the rate functional of the LDP and the $\Phi^4_3$ action functional coincide up to a constant. We wish to highlight that while our main result had previously been obtained by Barashkov (2022), the main focus of this work is on the approach through the stochastic quantisation equation.

math.PR↗

arXiv2023

Gaussian Rough Paths Lifts via Complementary Young Regularity

Inspired by recent advances in singular SPDE theory, we use the Poincar\'e inequality on Wiener space to show that controlled complementary Young regularity is sufficient to obtain Gaussian rough paths lifts. This allows us to completely bypass assumptions on the 2D variation regularity of the covariance and, as a consequence, we obtain cleaner proofs of approximation statements (with optimal convergence rates) and show the convergence of random Fourier series in rough paths metrics under minimal assumptions on the coefficients (which are sharper than those in the existent literature).

math.PR↗

arXiv2023

Perturbation theory for the $Φ^4_3$ measure, revisited with Hopf algebras

We give a relatively short, almost self-contained proof of the fact that the partition function of the suitably renormalised $Φ^4_3$ measure admits an asymptotic expansion, the coefficients of which converge as the ultraviolet cut-off is removed. We also examine the question of Borel summability of the asymptotic series. The proofs are based on Wiener chaos expansions, Hopf-algebraic methods, and bounds on the value of Feynman diagrams obtained through BPHZ renormalisation.

math-ph↗

arXiv2022

Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM

We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.

math.PR↗

arXiv2022

The constant coefficient in precise Laplace asymptotics for gPAM

This article resumes the analysis of precise Laplace asymptotics for the generalised Parabolic Anderson Model (gPAM) initiated by Peter Friz and the author. More precisely, we provide an explicit formula for the constant coefficient in the asymptotic expansion in terms of traces and Carleman-Fredholm determinants of certain explicit operators. The proof combines classical Gaussian analysis in abstract Wiener spaces with arguments from the theory of regularity structures. As an ingredient, we prove that the minimiser in the (extended) phase functional of gPAM has better than just Cameron-Martin regularity.

math.PR↗