Math & Stats Colloquium Series - Dr. Geordie Richards

Speaker: Dr. Geordie Richards

Talk Title: On the long-time statistical behavior of solutions to damped stochastic KdV

Abstract: 

The Korteweg–de Vries equation (KdV) is a nonlinear partial differential equation (PDE) that was derived in the 19th century to describe the evolution of shallow water waves and has since become the focus of a vast literature of research due to its rich mathematical structure and ubiquitous appearance in applications.  We will introduce the damped stochastic KdV, a stochastic PDE obtained by adding noise and a linear damping term to KdV.  Roughly speaking, the noise term models random environmental effects, and the damping is included to preserve equilibrium, meaning the existence of an invariant measure on the initial data phase space.  We will explain that, if this invariant measure is unique, it is necessarily ergodic, indicating that this measure encodes the long-time statistics of solutions to damped stochastic KdV.

With this background in place, we will review an asymptotic coupling strategy for proving uniqueness of the invariant measure for a given stochastic PDE.  This framework was developed in joint work with Nathan Glatt-Holtz (Indiana) and Jonathan Mattingly (Duke).  As a novel application, we will present a concise proof of uniqueness of the invariant measure for damped stochastic KdV, a result from joint work with Nathan Glatt-Holtz and Vincent Martinez (Hunter).  Such ergodic theorems are significant, as ergodicity is often assumed for deterministic evolution PDEs (e.g., when modeling turbulent flows), but any proof of this assumption appears beyond the reach of modern deterministic theory.  Here, the additive noise - introduced to reflect environmental influences - serendipitously enables a deeper understanding of the statistical behavior of solutions.