Publications
127 results found
Hairer M, Rosati T, 2024, Global existence for perturbations of the 2D stochastic Navier–Stokes equations with space-time white noise, Annals of PDE, Vol: 10, ISSN: 2524-5317
We prove global in time well-posedness for perturbations of the 2D stochastic Navier–Stokes equations ∂tu+u·∇u=Δu-∇p+ζ+ξ,u(0,·)=u0,div(u)=0, driven by additive space-time white noise ξ , with perturbation ζ in the Hölder–Besov space C-2+3κ , periodic boundary conditions and initial condition u∈ C-1+κ for any κ> 0 . The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a log –correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation ζ is not restricted to the Cameron–Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data u in L2 , the critical space of initial conditions.
Gerasimovičs A, Hairer M, Matetski K, 2024, Directed mean curvature flow in noisy environment, Communications on Pure and Applied Mathematics, Vol: 77, Pages: 1850-1939, ISSN: 0010-3640
We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.
Hairer M, Steele R, 2024, The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality, Archive for Rational Mechanics and Analysis, Vol: 248, ISSN: 0003-9527
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 al. in A diagram-free approach to the stochastic estimates in regularity structures, 2021. arXiv:2112.10739) in the multi-index setting, but our proof relies crucially on a novel version of the reconstruction theorem for a space of “pointed Besov modelled distributions”. As a consequence, the analytical core of the proof is quite short and self-contained, which should make it easier to adapt the proof to different contexts (such as the setting of discrete models).
Erhard D, Hairer M, 2023, A scaling limit of the parabolic Anderson model with exclusion interaction, COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, ISSN: 0010-3640
Hairer M, 2023, A STROLL AROUND THE CRITICAL POTTS MODEL, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, ISSN: 0273-0979
Hairer M, Le K, Rosati T, 2023, The Allen-Cahn equation with generic initial datum, PROBABILITY THEORY AND RELATED FIELDS, Vol: 186, Pages: 957-998, ISSN: 0178-8051
Cannizzaro G, Hairer M, 2023, The Brownian Web as a random R-tree, ELECTRONIC JOURNAL OF PROBABILITY, Vol: 28, ISSN: 1083-6489
Chandra A, Chevyrev I, Hairer M, et al., 2022, Langevin dynamic for the 2D Yang-Mills measure, Publications mathématiques de l'IHÉS, Vol: 136, Pages: 1-147, ISSN: 1618-1913
We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.
Gerencser M, Hairer M, 2022, Boundary renormalisation of SPDEs, COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, Vol: 47, Pages: 2070-2123, ISSN: 0360-5302
Hairer M, Li X-M, 2022, Generating diffusions with fractional Brownian motion, Communications in Mathematical Physics
We study fast / slow systems driven by a fractional Brownian motion $B$ withHurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic convergeson suitable timescales to a limiting Markov process and we describe itsgenerator. More precisely, if $Y^\varepsilon$ denotes a Markov process withsufficiently good mixing properties evolving on a fast timescale $\varepsilon\ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\;$$ converge to a regular diffusion without having to assume that $F$ averagesto $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar resultholds, but this time it does require $F$ to average to $0$. We also prove thatthe $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuousinterpolation between the homogenisation theorem for random ODEs with rapidlyoscillating right-hand sides ($H=1$) and the averaging of diffusion processes($H= \frac 12$).
Hairer M, Steele R, 2022, The $Φ_3^4$ measure has sub-Gaussian tails, Journal of Statistical Physics, ISSN: 0022-4715
We provide a very simple argument showing that the $\Phi^4_3$ measure doeshave quartic exponential tails, as expected from its formal expression. Thiscompletes the programme of recovering the verification of the Osterwalder andSchrader axioms for that measure based purely on SPDE techniques.
Hairer M, Schonbauer P, 2022, The support of singular stochastic partial differential equations, Forum of Mathematics, Pi, Vol: 10, Pages: 1-127, ISSN: 2050-5086
We obtain a generalisation of the Stroock–Varadhan support theorem for a large class of systems of subcritical singular stochastic partial differential equations driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup H of the renormalisation group such that any renormalisation procedure determines a unique coset g∘H . The support of the solution then depends only on this coset and is obtained by taking the closure of all solutions obtained by replacing the driving noises by smooth functions in the equation that is renormalised by some element of g∘H .One immediate corollary of our results is that the Φ43 measure in finite volume has full support, and the associated Langevin dynamic is exponentially ergodic.
Bruned Y, Gabriel F, Hairer M, et al., 2022, GEOMETRIC STOCHASTIC HEAT EQUATIONS, Publisher: AMER MATHEMATICAL SOC
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- Citations: 12
Hairer M, Schönbauer P, 2021, The support of singular stochastic PDEs, Publisher: Cambridge University Press
We obtain a generalisation of the Stroock-Varadhan support theorem for alarge class of systems of subcritical singular stochastic PDEs driven by anoise that is either white or approximately self-similar. The main problem thatwe face is the presence of renormalisation. In particular, it may happen ingeneral that different renormalisation procedures yield solutions withdifferent supports. One of the main steps in our construction is theidentification of a subgroup $\mathcal{H}$ of the renormalisation group suchthat any renormalisation procedure determines a unique coset$g\circ\mathcal{H}$. The support of the solution then only depends on thiscoset and is obtained by taking the closure of all solutions obtained byreplacing the driving noises by smooth functions in the equation that isrenormalised by some element of $g\circ\mathcal{H}$. One immediate corollary of our results is that the $\Phi^4_3$ measure infinite volume has full support and that the associated Langevin dynamic isexponentially ergodic.
Li X-M, Hairer M, 2021, Generating diffusions with fractional Brownian motion, Publisher: ArXiv
We study fast / slow systems driven by a fractional Brownian motion B with Hurst parameter H∈(13,1]. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if Yε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale ε≪1, the solutions of the equationdXε=ε12−HF(Xε,Yε)dB+F0(Xε,Yε)dtconverge to a regular diffusion without having to assume that F averages to 0, provided that H<12. For H>12, a similar result holds, but this time it does require F to average to 0. We also prove that the n-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides (H=1) and the averaging of diffusion processes (H=12).
Chandra A, Hairer M, Chevyrev I, et al., 2021, Renormalising SPDEs in regularity structures, Journal of the European Mathematical Society, Vol: 23, Pages: 869-947, 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, 2021, A noise-induced transition in the Lorenz system, Communications in Mathematical Physics, ISSN: 0010-3616
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.
Cannizzaro G, Hairer M, 2021, The Brownian Web as a random R-tree, Publisher: arXiv
Motivated by [G. Cannizzaro, M. Hairer, arXiv:2010.02766], we provide analternative characterisation of the Brownian Web (see [T\'oth B., Werner W.,Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M.Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescingBrownian motions starting from every point in $\mathbb R^2$ simultaneously, andfit it into the wider framework of random (spatial) $\mathbb R$-trees. Wedetermine some of its properties (e.g. its box-counting dimension) and recoversome which were determined in earlier works, such as duality, special pointsand convergence of the graphical representation of coalescing random walks.Along the way, we introduce a modification of the topology of spatial $\mathbbR$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05]and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes itPolish and could be of independent interest.
Cannizzaro G, Hairer M, 2021, The Brownian Castle, Publisher: arXiv
We introduce a $1+1$-dimensional temperature-dependent model such that theclassical ballistic deposition model is recovered as its zero-temperaturelimit. Its $\infty$-temperature version, which we refer to as the $0$-BallisticDeposition ($0$-BD) model, is a randomly evolving interface which, surprisinglyenough, does {\it not} belong to either the Edwards--Wilkinson (EW) or theKardar--Parisi--Zhang (KPZ) universality class. We show that $0$-BD has ascaling limit, a new stochastic process that we call {\it Brownian Castle} (BC)which, although it is "free", is distinct from EW and, like any otherrenormalisation fixed point, is scale-invariant, in this case under the $1:1:2$scaling (as opposed to $1:2:3$ for KPZ and $1:2:4$ for EW). In the presentarticle, we not only derive its finite-dimensional distributions, but alsoprovide a "global" construction of the Brownian Castle which has the advantageof highlighting the fact that it admits backward characteristics given by the(backward) Brownian Web (see [T\'oth B., Werner W., Probab. Theory RelatedFields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar,Ann. Probab., '04]). Among others, this characterisation enables us toestablish fine pathwise properties of BC and to relate these to special pointsof the Web. We prove that the Brownian Castle is a (strong) Markov and Fellerprocess on a suitable space of c\`adl\`ag functions and determine its long-timebehaviour. At last, we give a glimpse to its universality by proving theconvergence of $0$-BD to BC in a rather strong sense.
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.
Hairer M, Pardoux E, 2020, Fluctuations around a homogenised semilinear random PDE, Archive for Rational Mechanics and Analysis, Vol: 239, Pages: 151-217, ISSN: 0003-9527
We consider a semilinear parabolic partial differential equation in R+ × [0, 1]d, whered = 1, 2 or 3, with a highly oscillating random potential and either homogeneousDirichlet or Neumann boundary condition. If the amplitude of the oscillations has theright size compared to its typical spatiotemporal scale, then the solution of our equationconverges to the solution of a deterministic homogenised parabolic PDE, which is aform of law of large numbers. Our main interest is in the associated central limittheorem. Namely, we study the limit of a properly rescaled difference between theinitial random solution and its LLN limit. In dimension d = 1, that rescaled differenceconverges as one might expect to a centred Ornstein-Uhlenbeck process. However, indimension d = 2, the limit is a non-centred Gaussian process, while in dimensiond = 3, before taking the CLT limit, we need to subtract at an intermediate scale thesolution of a deterministic parabolic PDE, subject (in the case of Neumann boundarycondition) to a non-homogeneous Neumann boundary condition. Our proofs makeuse of the theory of regularity structures, in particular of the very recently developedmethodology allowing to treat parabolic PDEs with boundary conditions within thattheory.
Hairer M, Li X-M, 2020, Averaging dynamics driven by fractional Brownian motion, Annals of Probability, Vol: 48, Pages: 1826-1860, 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.
Bruned Y, Hairer M, Zambotti L, 2020, Renormalisation of Stochastic Partial Differential Equations, EMS Newsletter, Vol: 2020-3, Pages: 7-11, ISSN: 1027-488X
Friz PK, Hairer M, 2020, A Course on Rough Paths, Publisher: Springer International Publishing, ISBN: 9783030415556
Jaehnichen T, Koertner UHJ, 2019, Introduction, ZEITSCHRIFT FUR EVANGELISCHE ETHIK, Vol: 58, Pages: 1-2, ISSN: 0044-2674
Gerasimovics A, Hairer M, 2019, Hormander's theorem for semilinear SPDEs, Electronic Journal of Probability, Vol: 24, ISSN: 1083-6489
We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander’s bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris’s lemma to work directly on the Malliavin matrix, instead of the “reduced Malliavin matrix” which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs
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.
Flandoli F, Gubinelli M, Hairer M, 2019, Singular Random Dynamics Cetraro, Italy 2016 Introduction, SINGULAR RANDOM DYNAMICS: CETRARO, ITALY 2016, Editors: Flandoli, Gubinelli, Hairer, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 1-10, ISBN: 978-3-030-29544-8
Flandoli F, Gubinelli M, Hairer M, 2019, Singular Random Dynamics Cetraro, Italy 2016 Preface, SINGULAR RANDOM DYNAMICS: CETRARO, ITALY 2016, Editors: Flandoli, Gubinelli, Hairer, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: V-VII, ISBN: 978-3-030-29544-8
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