Title: Ergodicity and synchronization of the KPZ equation.
Abstract: The Kardar-Parisi-Zhang (KPZ) equation on the real line is well-known to have stationary distributions modulo additive constants given by Brownian motion with drift. In this talk, we will discuss some recent results which show that these distributions are totally ergodic and present progress toward the conjecture that these are the only ergodic stationary distributions of the KPZ equation. We study these problems through the lens of the synchronization by noise phenomenon and establish what is known as a one force — one solution principle: the solution to the KPZ equation (modulo an additive constant) started in the distant past from an initial condition with a given slope will converge almost surely to a Brownian motion with that drift for all but a random (possibly empty) countable set of slopes. The family of Brownian motions obtained in this way is known as the Busemann process for the equation and forms our main tool for studying these questions. Joint work with Firas Rassoul-Agha (Utah) and Timo Seppäläinen (Madison)
