Mathematics

 

The goal of this course is to present an approach to determine the large-scale universal behaviour of a family of stationary statistical mechanics systems at and beyond the critical dimension. These are systems for which scaling or Renormalization group arguments suggest a Gaussian limit for large space-time scales, with logarithmic corrections to diffusivity at the critical dimension, but whose rigorous proof has long eluded mathematicians. Examples to which the approach applies, and that we will discuss in the lectures, include random walks/diffusions in random environment, interacting particle systems and, especially, Stochastic Partial Differential Equations. In particular, the SPDEs we will be concerned with lie beyond the range of applicability of Hairer’s theory of regularity structures and are not expected to admit a local solution theory.

 

The approach finds its roots in the celebrated work of C. Kipnis and S.R.S. Varadhan (1986) and further developed over the last 40 years, by T. Komorowski, C. Landim, S. Olla, J. Quastel, H.T. Yau and many others. What they realised is that the large-scale behaviour of certain systems can be derived by identifying a suitable martingale approximation of so-called additive functionals of Markov processes, i.e. sums/integrals along their trajectories of suitable functionals of their state.

 

After recalling the basics of Markov processes and Kipnis-Varadhan theory, we will turn to more recent developments with a special focus on critical and supercritical SPDEs such as the Stochastic Burgers Equation, relevant in the description of driven diffusive systems, the Anisotropic KPZ equation, that arises in the context of random surface growth in 2 dimensions, and the stochastic Navier-Stokes equation, describing the evolution of the velocity field of turbulent flows.  

 

This course is part of the Probability Trimester at SISSA: https://sites.google.com/view/probsissa