A specialist in probability theory, stochastic processes and partial differential equations, Jeremy Quastel has been at the University of Toronto since 1998. He studied at McGill University, then the Courant Institute at New York University where he completed his PhD in 1990 under the direction of S.R.S. Varadhan; he has also worked at the Mathematical Sciences Research Institute in Berkeley, and UC Davis. His research is on the large scale behaviour of interacting particle systems and stochastic partial differential equations. He was a Sloan Fellow 1996–98, a Killam Fellow 2013–15, invited speaker at the International Congress of Mathematicians in Hyderabad 2010, gave the Current Developments in Mathematics 2011 and St. Flour 2012 lectures, and was a plenary speaker at the International Congress of Mathematical Physics in Aalborg 2012. Jeremy’s Medallion Lecture will be at JSM Montreal on August 5.
The Kardar-Parisi-Zhang equation and its universality class
Stochastic partial differential equations are used throughout the sciences to provide more realistic models than partial differential equations, taking into account the natural randomness or uncertainty in the environment. Sometimes the effects can be highly non-trivial, especially in small scale cases, e.g. biological cells, nanotechnology, chemical kinetics, but also large scale phenomena such as climate modelling. Our understanding of stochastic partial differential equations is still in a very primitive stage, and we are just starting to be able to study problems of genuine relevance to applications. A key scientific question for which there is no general recipe is how the input noise is transformed by a non-linear stochastic partial differential equation into fluctuations of the solution.
One of the most important non-linear stochastic partial differential equations is the Kardar-Parisi-Zhang equation (KPZ), introduced in 1986 as a canonical model for random surface growth. In the one dimensional case, it is equivalent to the stochastic Burgers equation, which is a model for randomly forced one-dimensional fluids. These models have been widely used in physics, but their mathematics was very poorly understood. At the physical level, it was discovered that the one-dimensional KPZ equation had highly non-trivial fluctuation behaviour, shared by a large collection of one-dimensional asymmetric, randomly forced systems: stochastic interface growth on a one dimensional substrate, randomly stirred one dimensional fluids, polymer chains directed in one dimension and fluctuating transversally in the other due to a random potential (with applications to domain interfaces in disordered crystals), driven lattice gas models, reaction-diffusion models in two-dimensional random media (including biological models such as bacterial colonies), randomly forced Hamilton-Jacobi equations, etc. These form the conjectural KPZ universality class. A combination of non-rigorous methods (renormalization, mode-coupling, replicas) and mathematical breakthroughs on a few special solvable models led to very precise predictions of universal scaling exponents and exact statistical distributions describing the long time properties. Surprisingly, they are the same as those found in random matrix theory: The Tracy-Widom distributions and their process level generalizations, the Airy processes.
These predictions have been repeatedly confirmed through Monte-Carlo simulation as well as experiments; in particular, recent spectacular experiments on turbulent liquid crystals. However, at the mathematical level the KPZ equation proved difficult, until recently, in a series of unexpected breakthroughs, the equation was shown to be well-posed, and exact distributions were computed for the main scaling invariant initial data. The goal of this talk will be to describe the background and the progress that has been made.