Stationary and shock solutions for the stochastic Burgers equation
The stochastic Burgers equation is a prototypical example of a conservation law with stochastic forcing. It is sometimes studied as a toy model for turbulence. Via the Cole–Hopf transform, it is also closely related to the KPZ equation, a model for the stochastic growth of a random surface. I will explain several recent results about the existence, classification, and properties of spacetime-stationary solutions to the stochastic Burgers equation on R^d for d≤3. These solutions represent the behavior of the model in large domains on long time scales. I will also describe a notion of stationary viscous shock solutions to the stochastic Burgers equation in d=1, and explain the relationship between such solutions and the KPZ problem. Finally, I will discuss extensions of some of these results to more general conservation laws. The talk is mostly based on joint work with Cole Graham and Lenya Ryzhik.