Publications
113 results found
Chandra A, Hairer M, Shen H, 2020, The dynamical sine-Gordon model in the full subcritical regime
We prove that the dynamical sine-Gordon equation on the two dimensional torusintroduced in [HS16] is locally well-posed for the entire subcritical regime.At first glance this equation is far out of the scope of the local existencetheory available in the framework of regularity structures [Hai14, BHZ16, CH16,BCCH17] since it involves a non-polynomial nonlinearity and the solution isexpected to be a distribution (without any additional small parameter as in[FG17, HX18]). In [HS16] this was overcome by a change of variable, but the newequation that arises has a multiplicative dependence on highly non-Gaussiannoises which makes stochastic estimates highly non-trivial - as a result [HS16]was only able to treat part of the subcritical regime. Moreover, the cumulantsof these noises fall out of the scope of the later work [CH16]. In this work wesystematically leverage "charge" cancellations specific to this model andobtain stochastic estimates that allow us to cover the entire subcriticalregime.
Chandra A, Hairer M, 2020, An analytic BPHZ theorem for regularity structures, Publisher: Being revised and resubmitted
We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Hölder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.
Chandra A, Hairer M, Chevyrev I, et al., 2020, Renormalising SPDEs in regularity structures, Journal of the European Mathematical Society, ISSN: 1435-9855
The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT. The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.
Zelati MC, Hairer M, 2020, A noise-induced transition in the Lorenz system
We consider a stochastic perturbation of the classical Lorenz system in therange of parameters for which the origin is the global attractor. We show thatadding noise in the last component causes a transition from a unique to exactlytwo ergodic invariant measures. The bifurcation threshold depends on thestrength of the noise: if the noise is weak, the only invariant measure isGaussian, while strong enough noise causes the appearance of a second ergodicinvariant measure.
Hairer M, Li X-M, Averaging dynamics driven by fractional Brownian motion, Annals of Probability, ISSN: 0091-1798
We consider slow / fast systems where the slow system is driven by fractionalBrownian motion with Hurst parameterH >12. We show that unlike in the caseH=12, convergence to the averaged solution takes place in probability and thelimiting process solves the ‘naïvely’ averaged equation. Our proof strongly relieson the recently obtained stochastic sewing lemma.
Gerencsér M, Hairer M, 2019, A solution theory for quasilinear singular SPDEs, Communications on Pure and Applied Mathematics, Vol: 72, Pages: 1983-2005, ISSN: 0010-3640
We give a construction allowing us to build local renormalized solutions to general quasilinear stochastic PDEs within the theory of regularity structures, thus greatly generalizing the recent results of [1, 5, 11]. Loosely speaking, our construction covers quasilinear variants of all classes of equations for which the general construction of [3, 4, 7] applies, including in particular one‐dimensional systems with KPZ‐type nonlinearities driven by space‐time white noise. In a less singular and more specific case, we furthermore show that the counterterms introduced by the renormalization procedure are given by local functionals of the solution. The main feature of our construction is that it allows exploitation of a number of existing results developed for the semilinear case, so that the number of additional arguments it requires is relatively small.
Hairer M, Li X-M, 2019, Averaging dynamics driven by fractional Brownian motion
We consider slow / fast systems where the slow system is driven by fractionalBrownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in thecase $H={1\over 2}$, convergence to the averaged solution takes place inprobability and the limiting process solves the 'na\"ively' averaged equation.Our proof strongly relies on the recently obtained stochastic sewing lemma.
Gerasimovics A, Hairer M, 2019, Hormander's theorem for semilinear SPDEs, Publisher: UNIV WASHINGTON, DEPT MATHEMATICS
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Bruned Y, Hairer M, Zambotti L, 2018, Algebraic renormalisation of regularity structures, Inventiones Mathematicae, ISSN: 0020-9910
We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations. Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.
Hairer M, Quastel J, 2018, A class of growth models rescaling to KPZ, Forum of Mathematics, Pi, Vol: 6, ISSN: 2050-5086
We consider a large class of 1+1-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf-Cole solutions to the KPZ equation.
Hairer M, 2018, Renormalisation of parabolic stochastic PDEs, Japanese Journal of Mathematics, Vol: 13, Pages: 187-233, ISSN: 0289-2316
We give a survey of recent result regarding scaling limits of systems from statistical mechanics, as well as the universality of the behaviour of such systems in so-called cross-over regimes. It transpires that some of these universal objects are described by singular stochastic PDEs. We then give a survey of the recently developed theory of regularity structures which allows to build these objects and to describe some of their properties. We place particular emphasis on the renormalisation procedure required to give meaning to these equations.These are expanded notes of the 20th Takagi Lectures held at The University of Tokyo on November 4, 2017.
Hairer M, Xu W, 2018, Large-scale limit of interface fluctuation models
We extend the weak universality of KPZ in [Hairer-Quastel] to weakly asymmetric interface models with general growth mechanisms beyond polynomials. A key new ingredient is a pointwise bound on correlations of trigonometric functions of Gaussians in terms of their polynomial counterparts. This enables us to reduce the problem of a general nonlinearity with sufficient regularity to that of a polynomial.
Hairer M, Mattingly J, 2018, The strong Feller property for singular stochastic PDEs, ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, Vol: 54, Pages: 1314-1340, ISSN: 0246-0203
Hairer M, 2018, An analyst's take on the BPHZ theorem, Publisher: arXiv
We provide a self-contained formulation of the BPHZ theorem in the Euclidean context, which yields a systematic procedure to "renormalise" otherwise divergent integrals appearing in generalised convolutions of functions with a singularity of prescribed order at their origin. We hope that the formulation given in this article will appeal to an analytically minded audience and that it will help to clarify to what extent such renormalisations are arbitrary (or not). In particular, we do not assume any background whatsoever in quantum field theory and we stay away from any discussion of the physical context in which such problems typically arise.
Cuneo N, Eckmann J-P, Hairer M, et al., 2018, Non-equilibrium steady states for networks of oscillators, Electronic Journal of Probability, Vol: 23, ISSN: 1083-6489
Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies. In this paper, we show how some of the results extend to more complicated networks. We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.
Hairer M, Matetski K, 2018, DISCRETISATIONS OF ROUGH STOCHASTIC PDES, ANNALS OF PROBABILITY, Vol: 46, Pages: 1651-1709, ISSN: 0091-1798
Hairer M, Iberti M, 2018, Tightness of the Ising-Kac model on the two-dimensional torus, Journal of Statistical Physics, Vol: 171, Pages: 632-655, ISSN: 1572-9613
We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by Mourrat and Weber (Commun Pure Appl Math 70:717–812, 2017) and a method by Tsatsoulis and Weber employed in (arXiv:1609.08447 2016), we show tightness for the sequence of Gibbs measures of the Ising–Kac model near criticality and characterise the law of the limit as the Φ42 measure on the torus. Our result is very similar to the one obtained by Cassandro et al. (J Stat Phys 78(3):1131–1138, 1995) on Z2 , but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature/large mass/small coupling assumption present in earlier results.
Hairer M, Labbé C, 2018, Multiplicative stochastic heat equations on the whole space, Journal of the European Mathematical Society, Vol: 20, Pages: 1005-1054, ISSN: 1435-9855
We carry out the construction of some ill-posed multiplicative stochastic heat equations on unbounded domains. The two main equations our result covers are, on the one hand the parabolic Anderson model on R³, and on the other hand the KPZ equation on R via the Cole-Hopf transform. To perform these constructions, we adapt the theory of regularity structures to the setting of weighted Besov spaces. One particular feature of our construction is that it allows one to start both equations from a Dirac mass at the initial time.
Hairer M, Iyer G, Koralov L, et al., 2018, A FRACTIONAL KINETIC PROCESS DESCRIBING THE INTERMEDIATE TIME BEHAVIOUR OF CELLULAR FLOWS, ANNALS OF PROBABILITY, Vol: 46, Pages: 897-955, ISSN: 0091-1798
Gerencsér M, Hairer M, 2018, Singular SPDEs in domains with boundaries, Probability Theory and Related Fields, Pages: 1-62, ISSN: 0178-8051
© 2018 The Author(s) We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition.
Bruned Y, Chandra A, Chevyrev I, et al., 2018, Renormalising SPDEs in regularity structures
The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear stochastic PDEs. Under very mild additional assumptions, it was then shown in arXiv:1612.08138 that large classes of driving noises exhibiting the relevant small-scale behaviour can be lifted to such a regularity structure in a robust way, following a renormalisation procedure reminiscent of the BPHZ procedure arising in perturbative QFT.The present work completes this programme by constructing an action of the renormalisation group onto a suitable class of stochastic PDEs which is intertwined with its action on the corresponding space of models. This shows in particular that solutions constructed from the BPHZ lift of a smooth driving noise coincide with the classical solutions of a modified PDE. This yields a very general black box type local existence and stability theorem for a wide class of singular nonlinear SPDEs.
Hairer M, Xu W, 2018, Large-scale behavior of three-dimensional continuous phase coexistence models, Communications on Pure and Applied Mathematics, Vol: 71, Pages: 688-746, ISSN: 0010-3640
We study a class of three-dimensional continuous phase coexistence models, and show that, under different symmetry assumptions on the potential, the large-scale behavior of such models near a bifurcation point is described by the dynamical Φp₃ models for p ∊ {2,3,4}. This result is specific to space dimension 3 and does not hold in dimension 2.
Bruned Y, Hairer M, Zambotti L, 2018, Algebraic renormalisation of regularity structures
We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms. This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context. This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations.Our construction is based on bialgebras of decorated coloured forests in cointeraction. More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory. Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory.
Hairer M, Shen H, 2017, A central limit theorem for the KPZ equation, Annals of Probability, Vol: 45, Pages: 4167-4221, ISSN: 0091-1798
We consider the KPZ equation in one space dimension driven by a stationary centred space–time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf–Cole solution to the KPZ equation driven by space–time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.
Hairer M, Labbe C, 2017, The reconstruction theorem in Besov spaces, Journal of Functional Analysis, Vol: 273, Pages: 2578-2618, ISSN: 0022-1236
The theory of regularity structures [9] sets up an abstract framework of modelled distributions generalising the usual Hölder functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of modelled distributions into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces Bp,qγ with non-integer regularity indices. We then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and Schauder-type estimates.
Erhard D, Hairer M, 2017, Discretisation of regularity structures
We introduce a general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs. The approach pursued in this article is that we do not focus on any one specific discretisation procedure. Instead, we assume that we are given a scale ε>0 and a "black box" describing the behaviour of our discretised objects at scales below ε.
Hairer M, Koralov L, Pajor-Gyulai Z, 2016, From averaging to homogenization in cellular flows - An exact description of the transition, ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, Vol: 52, Pages: 1592-1613, ISSN: 0246-0203
Hairer M, Matetski K, 2015, Optimal rate of convergence for stochastic Burgers-type equations, Stochastics and Partial Differential Equations: Analysis and Computations, Vol: 4, Pages: 402-437, ISSN: 2194-0401
Recently, a solution theory for one-dimensional stochastic PDEs of Burgers type driven by space-time white noise was developed. In particular, it was shown that natural numerical approximations of these equations converge and that their convergence rate in the uniform topology is arbitrarily close to 1616. In the present article we improve this result in the case of additive noise by proving that the optimal rate of convergence is arbitrarily close to 1212.
Hairer M, Shen H, 2015, The Dynamical Sine-Gordon Model, Communications in Mathematical Physics, Vol: 341, Pages: 933-989, ISSN: 0010-3616
We introduce the dynamical sine-Gordon equation in two space dimensions with parameter ββ, which is the natural dynamic associated to the usual quantum sine-Gordon model. It is shown that when β2∈(0,16π3)β2∈(0,16π3) the Wick renormalised equation is well-posed. In the regime β2∈(0,4π)β2∈(0,4π), the Da Prato–Debussche method [J Funct Anal 196(1):180–210, 2002; Ann Probab 31(4):1900–1916, 2003] applies, while for β2∈[4π,16π3)β2∈[4π,16π3), the solution theory is provided via the theory of regularity structures [Hairer, Invent Math 198(2):269–504, 2014]. We also show that this model arises naturally from a class of 2+12+1 -dimensional equilibrium interface fluctuation models with periodic nonlinearities. The main mathematical difficulty arises in the construction of the model for the associated regularity structure where the role of the noise is played by a non-Gaussian random distribution similar to the complex multiplicative Gaussian chaos recently analysed in Lacoin et al. [Commun Math Phys 337(2):569–632, 2015].
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