Martin Hairer is a mathematician whose work joins probability theory and analysis. He developed regularity structures to give precise meaning to classes of singular stochastic partial differential equations. His EPFL biography identifies this work as central to his 2014 Fields Medal.[ref]
https://www.epfl.ch/labs/propde/prof-martin-hairer/. Institutional biography and principal publications
The motivating obstacle is that a noisy solution can be too irregular for the nonlinear operations appearing in its own equation. Smoothing the input makes an ordinary equation possible, but the resulting solutions need not converge when the smoothing is removed. Regularity structures replace ordinary polynomial local expansions with expansions tailored to the singular input and organize the necessary corrections. The theory proves convergence of appropriate corrected approximations for the equations it covers.[ref]
A theory of regularity structures by Martin Hairer. Abstract and Introduction: local expansions, nonlinear operations and renormalized approximation
His An Introduction to Stochastic PDEs by Martin Hairer gives a complementary starting point through the stochastic heat equation and mild solutions. Integrating the heat evolution against a random driving term clarifies why solving a noisy equation requires specifying both the evolution operator and the interpretation of the noise.[ref]
An Introduction to Stochastic PDEs by Martin Hairer. Sections 2 and 5: motivating equations and stochastic convolution
Articles by others on the same topic
Martin Hairer is an Austrian mathematician known for his work in the field of stochastic analysis, particularly in relation to the theory of stochastic partial differential equations (SPDEs). He is recognized for developing a mathematical framework known as the theory of regularity structures, which provides tools to analyze and solve SPDEs that arise in various applications, including mathematical physics.