MARK KAC SEMINAR

Mass-stationarity means that the origin is at a typical location in the mass of a random measure. For a simple example, consider the stationary Poisson process on the line conditioned on having a point at the origin. The origin is then at a typical point (at a typical location in the mass) because shifting the origin to the n:th point on the right (or on the left) does not alter the fact that the inter-point distances are i.i.d. exponential. Another (less obvious) example is the local time at zero of a two-sided standard Brownian motion.

In the first part of the talk we shall focus on the line, starting with intriguing observations such as Liggett's extra-head scheme and how to find the Brownian bridge in the path of a Brownian motion. We then consider the shift-coupling problem of how to shift the origin from a typical location in the mass of one random measure to a typical location in the mass of another random measure.

In the second part of the talk we extend the view beyond the line, moving through the Poisson process in the plane and d-dimesional space to a general Palm theory of mass-stationarity and shift-coupling on groups.

The KPZ equation was introduced in the '80s by Kardar, Parisi and Zhang as a simplified model for stochastic growth of one-dimensional interfaces. From heuristic considerations, it has been conjectured that scaling limits of general stationary, non-equilibrium fluctuations of one-dimensional, conservative dynamics are governed by the KPZ equation (the so-called weak KPZ universality conjecture). However, from the mathematical point of view, the KPZ equation is not well posed, and more than 25 years passed until a satisfactory notion of solution was built.

In fact, the celebrated Hairer's theory of regularity structures appeared as a way to make sense of the KPZ equation. An earlier, weaker notion of ‘energy solutions’ of the KPZ equation was introduced in joint works with Massimiliano Gubinelli and Patricia Gonçalves. Very recently, Gubinelli and Perkowski have proved the uniqueness of stationary solutions of the KPZ equation, paving the way to the proof of the weak KPZ universality conjecture.

The aim of this series of lectures is to present a complete proof of the weak KPZ universality conjecture, using the notion of energy solutions of the KPZ equation.