Title: Stationary solutions for 1D stochastic Burgers and KPZ universality
Abstract: In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties. We will classify all the ergodic components, establish the “one force---one solution” principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic results have their counterparts in the geodesic/polymer language. These objects are also believed to be in the KPZ universality class.
In the second part, we will study certain renormalization action on random point fields of concentration and separation, which arises naturally in the context of the stochastic Burgers equation. There are two goals: one is to establish the stability of the fixed point, which leads to a convergence rate estimate of the stochastic Burgers equation, and the other is to provide a geometric viewpoint of the KPZ 1:2:3 scaling and to potentially enlarge the universality class. Some preliminary results are given in both directions.
