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Professor Martin Hairer
2015
Exchanges
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Professor Martin Hairer was one of four recipients of the 2014 Fields Medal, widely viewed as the highest honour a mathematician can receive. ...
He has enjoyed great success communicating mathematics to a range of audiences and has also developed music editing software.In this interview, early career mathematicians, Dr Martine Barons (MJB) and ...
About the Authors Figure 1 . 1 Martin Hairer in 2014, portrait via The Royal Society Barons & Chleboun. ...
doi:10.31273/eirj.v3i1.123
fatcat:iv4bxoepmnevxbwem3fppdmxau
Szorstkie trajektorie i struktury regularności., Martin Hairer nagrodzony medalem Fieldsa
2015
Wiadomosci Matematyczne
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Martin ma zainteresowania pozamatematyczne. ...
W odczycie w Institute for Advanced Study w Princeton, Martin powiedział: "zaintrygowało mnie pytanie, jak można wyprowadzić te równania, które nie mają żadnego sensu". ...
doi:10.14708/wm.v51i1.4264
fatcat:hene6am5grhl5k5pku52njbziy
"Da fühle ich mich manchmal schon wie eine Art Ingenieur" – Martin Hairer im Interview
2014
Mitteilungen der DMV
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MDMV 22 / 2014 | 209-211
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Martin Hairer (Foto: IMU/ICM)
Und wie arbeiten Sie konkret? ...
Seit 2014 ist Martin Hairer auch Fellow der Royal Society. Er lehrt heute auf einer vom britischen Königshaus gestifteten ("Regius") Professur für Mathematik an der Universität von Warwick. ...
doi:10.1515/dmvm-2014-0084
fatcat:g66db4kt4zc7ni6ie5bj2zascm
Rough Stochastic PDEs
[article]
2010
arXiv
pre-print
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Remark 5.4 A canonical choice of A and H is to take for H the Cameron-Martin space H ν of ν and for A the restriction operator (with domain H ν ∈ B). ...
arXiv:1008.1708v1
fatcat:34bpa4wpwjbn5pc2mpamfe5ogm
Singular stochastic PDEs
[article]
2014
arXiv
pre-print
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We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This problem can be addressed within the framework of the recently developed theory of "regularity structures", which allows to describe candidate solutions locally by a "jet", but where the usual Taylor polynomials are replaced by a sequence of custom-built
arXiv:1403.6353v1
fatcat:6bcwmdooabaldkxqjdfex7lmim
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... s. In order to illustrate the theory, we focus on the particular example of the Kardar-Parisi-Zhang equation, a popular model for interface propagation.
Hypoellipticity in Infinite Dimensions
[article]
2009
arXiv
pre-print
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We consider semilinear parabolic stochastic PDEs driven by additive noise. The question addressed in this note is that of the regularity of transition probabilities. If the equation satisfies a Hormander 'bracket condition', then any finite-dimensional projection of the solution has a smooth density with respect to Lebesgue measure. One key ingredient in the argument is a bound on 'Wiener polynomials' that plays a role analogue to Norris' lemma.
arXiv:0910.0315v1
fatcat:5u4q6hlf3rbyfpz7vchh7qqb5e
The Brownian fan
[article]
2014
arXiv
pre-print
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We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article DMC. DMC is a simulation technique that uses branching particle systems to represent expectations associated with Feynman-Kac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to
arXiv:1404.2928v1
fatcat:e7zpnopx4fbhnkz2z5anean43y
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... nverge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naïve" generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan", is a very natural new mathematical object of independent interest.
Boundary renormalisation of SPDEs
[article]
2021
arXiv
pre-print
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We consider the continuum parabolic Anderson model (PAM) and the dynamical Φ^4 equation on the 3-dimensional cube with boundary conditions. While the Dirichlet solution theories are relatively standard, the case of Neumann/Robin boundary conditions gives rise to a divergent boundary renormalisation. Furthermore for Φ^4_3 a 'boundary triviality' result is obtained: if one approximates the equation with Neumann boundary conditions and the usual bulk renormalisation, then the limiting process
arXiv:2110.03656v1
fatcat:7a5etsx3yfexjiieq7usrk7o5q
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... ides with the one obtained using Dirichlet boundary conditions.
Discretisation of regularity structures
[article]
2017
arXiv
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This methodology was for instance successfully applied by Hairer and Matetski [HM ] to prove that the Φ 4 3 -measure built in [GJ , Fel ] is invariant for the Φ 4 3 -equation. ...
We note at this point that following the approach of Hairer and Labbé [HL ] one could probably also deal with non-compact situations. ...
arXiv:1705.02836v1
fatcat:hmzd3vmfqbgjjghzifygkrdu5q
Renormalisation of parabolic stochastic PDEs
[article]
2018
arXiv
pre-print
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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
arXiv:1803.03044v1
fatcat:23utoi3zorfybbmx3s6oj4tjee
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... edure required to give meaning to these equations. These are expanded notes of the 20th Takagi lectures held at Tokyo University on November 4, 2017.
An Introduction to Stochastic PDEs
[article]
2009
arXiv
pre-print
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Let us discuss a few properties of the Cameron-Martin space. ...
3.41 (Cameron-Martin) For h ∈ B, define the map T h : B → B by T h (x) = x + h. ...
space ofμ = A * µ is given by the image under A of the Cameron-Martin space of µ. ...
arXiv:0907.4178v1
fatcat:ajgak4rufnaxrp4fiqag4zhg5a
On Malliavin's proof of Hörmander's theorem
[article]
2011
arXiv
pre-print
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The aim of this note is to provide a short and self-contained proof of Hörmander's theorem about the smoothness of transition probabilities for a diffusion under Hörmander's "brackets condition". While both the result and the technique of proof are well-known, the exposition given here is novel in two aspects. First, we introduce Malliavin calculus in an "intuitive" way, without using Wiener's chaos decomposition. While this may make it difficult to prove some of the standard results in
arXiv:1103.1998v1
fatcat:6k3pgvnolzagrk2xln2c4cje2m
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... n calculus (boundedness of the derivative operator in L^p spaces for example), we are able to bypass these and to replace them by weaker results that are still sufficient for our purpose. Second, we introduce a notion of "almost implication" and "almost truth" (somewhat similar to what is done in fuzzy logic) which allows, once the foundations of Malliavin calculus are laid out, to give a very short and streamlined proof of Hörmader's theorem that focuses on the main ideas without clouding it by technical details.
Stochastic PDEs with multiscale structure
[article]
2012
arXiv
pre-print
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We study the spatial homogenisation of parabolic linear stochastic PDEs exhibiting a two-scale structure both at the level of the linear operator and at the level of the Gaussian driving noise. We show that in some cases, in particular when the forcing is given by space-time white noise, it may happen that the homogenised SPDE is not what one would expect from existing results for PDEs with more regular forcing terms.
arXiv:1202.1775v1
fatcat:usqtvgu5w5he5kiqxx2wamzmei
Periodic homogenization with an interface
[article]
2009
arXiv
pre-print
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We consider a diffusion process with coefficients that are periodic outside of an 'interface region' of finite thickness. The question investigated in the articles [1,2] is the limiting long time / large scale behaviour of such a process under diffusive rescaling. It is clear that outside of the interface, the limiting process must behave like Brownian motion, with diffusion matrices given by the standard theory of homogenization. The interesting behaviour therefore occurs on the interface. Our
arXiv:0910.0314v1
fatcat:c6sa35oavbavzf4ryuf6auwrnq
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... main result is that the limiting process is a semimartingale whose bounded variation part is proportional to the local time spent on the interface. We also exhibit an explicit way of identifying its parameters in terms of the coefficients of the original diffusion. Our method of proof relies on the framework provided by Freidlin and Wentzell for diffusion processes on a graph in order to identify the generator of the limiting process.
The Brownian Castle
[article]
2021
arXiv
pre-print
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We introduce a 1+1-dimensional temperature-dependent model such that the classical ballistic deposition model is recovered as its zero-temperature limit. Its ∞-temperature version, which we refer to as the 0-Ballistic Deposition (0-BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards–Wilkinson (EW) or the Kardar–Parisi–Zhang (KPZ) universality class. We show that 0-BD has a scaling limit, a new stochastic process that we call Brownian
arXiv:2010.02766v2
fatcat:cj2tyhkedrf5dfhso4ls5jar54
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... le (BC) which, although it is "free", is distinct from EW and, like any other renormalisation 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 present article, we not only derive its finite-dimensional distributions, but also provide a "global" construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [Tóth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long-time behaviour. At last, we give a glimpse to its universality by proving the convergence of 0-BD to BC in a rather strong sense.
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