## About the work of Martin Hairer

The 2014 Fields medals were awarded to Artur Avila, Manjul Bhargava, Martin Hairer and Maryam Mirzakhani. Their works are shortly described in the IMU announcement. Artur Avila main contributions are in ergodic theory and dynamical systems. Manjul Bhargava’s are in number theory. Martin Hairer’s are in stochastic analysis and Maryam Mirzakhani’s are in hyperbolic geometry.

Maryam Mirzakhani is the first woman to receive the Fields medal. This is certainly an historic event and great news for a discipline in which men are traditionally overrepresented.

In this short post, I would like to discuss, at a non-technical level, some ideas related to the beautiful works of Martin Hairer, whom I know since quite a long time and whose works are the most familiar to me among the medalists. Hairer’s recent results deal with the study of very rough random dynamical systems and build on ideas going back at least to Kiyoshi Ito which have been revisited more recently by Terry Lyons.

Assume that we are interested in defining solutions for a differential equation that writes

$y(t)=y(0)+\int_0^t \sigma(y(s)) dx(s)$

where $\sigma: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ and $x:[0,+\infty)\to \mathbb{R}^n$ is the driving path of the equation. If $x$ and $\sigma$ are regular enough, let us say smooth, then by using a fixed point argument, we can prove that the equation has a unique locally defined solution $y$. The integral $\int_0^t \sigma(y(s)) dx(s)$ is then understood in the classical Riemann-Stieltjes sense.

Assume now that we would like to understand what could be a solution of the same equation when $x$ is not regular anymore, but  let us say has a bounded $p$-variation with $p > 1$. A natural idea is to consider a sequence $x_n$ of smooth approximations of $x$, look at the equation

$y_n(t)=y(0)+\int_0^t \sigma(y_n(s)) dx_n(s)$

and hope for the best, which is the convergence of $y_n$ to some $y$ which would be universal, that is, independent from the actually approximating sequence $x_n$. This idea has been carried out in the 1990’s by Terry Lyons in his theory of rough paths (a set of lecture notes on the theory are available on the blog). A key insight is the correct topology in which we have to understand the statement $x_n$ approximates $x$. Terry Lyons proved that this topology depends on the integer part of $p$. If $1 \le p <2$, then it is enough that $x_n\to x$ in the $p$-variation topology. If $2 \le p<3$, it is not enough that $x_n\to x$, we also want the convergence in $p$-variation of the double integrals $\int x^i_n dx^j_n$. If $3\le p <4$ we will also require a convergence in $p$-variation of the triple integrals, and so on and so forth. Thus, the topology that works is intimately related to the level of irregularity of the path $x$.

Martin Hairer recently developed a theory of regularity structures that encompasses as a special case the rough paths theory of Terry Lyons and that somehow can be thought as an extension of Lyons’ ideas to the space variables . His theory allows to give a sense to solutions of extremely rough partial differential equations that naturally arise in mathematical physics. A primary example that Martin Hairer was able to deal with is the Kardar-Parisi-Zhang equation (in short KPZ) that I now shortly discuss. The (one-dimensional) KPZ equation writes

$\partial_t h =\partial_x^2 h + (\partial_x h)^2 -C+\xi$

where $\xi$ is a white noise. From classical estimates, we expect $h$ to look like a Brownian motion in space at any fixed time. The derivative $\partial_x h$ is then a distribution, so the square of $\partial_x h$ does not make sense !

Similarly to the rough paths approach, Martin Hairer proves that if we consider a sequence of equations

$\partial_t h_\varepsilon =\partial_x^2 h_\varepsilon + (\partial_x h_\varepsilon)^2 -C_\varepsilon+\xi_\varepsilon$

where $\xi_\varepsilon$ is smooth and converges in an appropriate sense to the white noise $\xi$ and where $C_\varepsilon$ is a well-chosen constant, then the solution $h_\varepsilon$ converges to a limit which is universal. A key insight is to reduce the problem of the convergence of $h_\varepsilon$ to the problem of the convergence of a finite set of “building blocks” of the approximation. By analogy with the rough paths theory, these building blocks may be thought as an analogue of the iterated integrals of the driving path. As in the rough paths case, the topology that works is intimately related to the level of irregularity of the noise $\xi$.

The regularity structure of Martin Hairer applies far beyond the one-dimensional KPZ equation and proposes actually a general set of tools to explicitly construct good approximations given a singular equation. This is a revolutionary approach that offers a new look on several fundamental equations in mathematical physics that were long thought to be impossible to handle in a rigorous mathematical way.

Here is the laudation of Hairer’s work by Ofer Zeitouni which is followed by Martin Hairer’s lecture.

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