Abstract:
We study the one-dimensional stochastic Burgers equation on the real line, forced by a white-in-time, color-in-space homogeneous noise, with either positive or zero viscosity. Under mild assumptions on the forcing, we establish that for every fixed average velocity, a unique ergodic solution exists and the One Force --- One Solution principle holds. Moreover, in the inviscid limit the positive viscosity ergodic solutions will converge to the zero-viscosity ones. The analysis utilizes the connection between the Burgers equation and the associated variational problems and direct polymers models, where the ergodic solution problem can be reformulated in terms of the existence of the infinite-volume and zero-temperature limits of the polymer measures. Finally, we will discuss the generalizations and challenges in higher dimensions and for general non-quadratic Hamiltonians, as well as the connection to KPZ universality problems.
Biography:
Liying Li received his Ph.D. from the Courant Institute at New York University in 2019. He subsequently held postdoctoral positions at the University of Toronto and the University of Michigan during 2019-2023. He is currently an associate professor at the Southern University of Science and Technology. His primary research area is probability, with a specific focus on the ergodic properties of SPDEs and models associated with the Kardar--Parisi--Zhang universality class, including stochastic Burgers/Hamilton--Jacobi equations, directed polymers, first- and last-passage percolation and so on.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai
This event is open to the NYU Shanghai community and Math community.