Title: Invariant measures and shock fluctuations in the Kardar-Parisi-Zhang equation
Abstract: The Kardari-Parisi-Zhang (KPZ) equation is a stochastic PDE that was first introduced in the physics literature in 1986. It is used as a model of growing interfaces, such as a flame front, the boundary of a liquid crystal, or the boundary of a bacteria colony. An invariant measure for the KPZ equation is a measure on the space of continuous functions, so that, when the initial data for the equation is distributed according to that measure, the recentered solution has the same distribution for all positive times. In one space and one time dimension, when the spatial domain of the KPZ equation is the real line, it is known that Brownian motion with any choice of drift parameter is invariant. It has been conjectured that these constitute all extremal invariant measures for the KPZ equation. Recently, Janjigian, Rassoul-Agha, and Seppäläinen reduced this conjecture to proving the nonexistence of invariant measures supported on functions having asymptotic slopes +u and -u at + and - infinity, respectively, for some u > 0. Such functions asymptotically have the shape of the capital letter "V," and so we call the associated measures "V-shaped invariant measures." In this talk, I will discuss a recent preprint, joint with Alex Dunlap, where we complete the classification of the extremal invariant measures for the KPZ equation by ruling out the existence of V-shaped invariant measures. To solve this problem, we study the random fluctuations of shocks in the KPZ equation started from some special initial conditions. This talk will give a gentle overview for a general audience; no prior knowledge of stochastic PDEs, Brownian motion, or Markov processes will be assumed.