Chercheurs

Mathématiques / HAIRER, MARTIN

M.Hairer

Position

Regius Professor of Mathematics at University of Warwick, UK

Thèmes de recherche

Stochastic PDE\'s, stochastic analysis, functional analysis, homogenisation theory

A mathematical problem that is important throughout science is to understand the influence of noise on differential equations, and on the long time behavior of the solutions. This problem was solved for ordinary differential equations by Itó in the 1940s. For partial differential equations, a comprehensive theory has proved to be more elusive, and only particular cases (linear equations, tame nonlinearities, etc.) had been treated satisfactorily.

Hairer’s work addresses two central aspects of the theory. Together with Mattingly he employed the Malliavin calculus along with new methods to establish the ergodicity of the two-dimensional stochastic Navier-Stokes equation.

Building on the rough-path approach of Lyons for stochastic ordinary differential equations, Hairer then created an abstract theory of regularity structures for stochastic partial differential equations (SPDEs). This allows Taylor-like expansions around any point in space and time. The new theory allowed him to construct systematically solutions to singular non-linear SPDEs as fixed points of a renormalization procedure.

Hairer was thus able to give, for the first time, a rigorous intrinsic meaning to many SPDEs arising in physics.

Prix et récompenses

2014 Fields Medal for his outstanding contributions to the theory of stochastic partial differential equations, and in particular created a theory of regularity structures for such equations.

Recipient of a 2008 Whitehead prize awarded by the LMS
• Recipient of a 2008 Philip Leverhulme Prize
• Recipient of a 2009 Royal Society Wolfson Research Merit Award
• Member of the scientific steering committee of the IHP (2012-present)
• Member of the scientific steering committee of the Oberwolfach Institute (2013-present)
• Member of the scientific steering committee of ETHZ-ITS (2013-present)
• Two works were the subject of a “Seminaire Bourbaki” (2009 by Kupiainen and 2013 ´
by Zambotti)
• Laureate of the 2013 Fermat prize, shared with C. De Lellis.
• Fellow of the Royal Society (since 2014).

Publications

Hairer M, Exponential Mixing Properties of Stochastic PDEs through asymptotic coupling. Probab. Theory Rel. Fields, 124(2002), pp. 345–380

Eckmann J.-P and Hairer M, Spectral Properties of Hypoelliptic Operators. Commun. Math. Phys., 235 (2003), pp. 233–253

Hairer M and Mattingly J, Ergodicity of the 2D Navier-Stokes Equations with degenerate Stochastic Forcing. Annals of Maths, 164 (2006), pp. 993–1032

Hairer M and Ohashi A, Ergodic theory for SDEs with extrinsic memory. Annals of Probability, 35 (2007), pp. 1950–1977

Hairer M, How hot can a heat bath get? Commun. Math. Phys., 292 (2009), pp 131–177

Hairer, M, Mattingly, JC, and Scheutzow, M, Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probability Theory and Related Fields 149 (2011), pp 223-259

Hairer M, Rough Stochastic PDEs. Commun. Pure Appl. Math., 64 (2011), no 11, pp. 1547–1585

Hairer M, Solving the KPZ equation. Annals of Maths, 178 (2013), no 2, pp. 559-664