Burgers turbulence, Chernoff's distribution, complete integrability and stochastic coalescence

Series: 
PDE Seminar
Tuesday, February 24, 2009 - 15:05
1.5 hours (actually 80 minutes)
Location: 
Skiles 255
,  
Brown University
The problem of understanding the parabolic hull of Brownian motion arises in two different fields. In mathematical physics this is the Burgers-Hopf caricature of turbulence (very interesting, even if not entirely turbulent). In statistics, the limit distribution we study was first considered by Chernoff, and forms the cornerstone of a large class of limit theorems that have now come to be called 'cube-root-asymptotics'. It was in the statistical context that the problem was first solved completely in the mid-80s by Groeneboom in a tour de force of hard analysis. We consider another approach to his solution motivated by recent work on stochastic coalescence (especially work of Duchon, Bertoin, and my joint work with Bob Pego). The virtues of this approach are simplicity, generality, and the appearance of a completely unexpected Lax pair. If time permits, I will also indicate some tantalizing links of this approach with random matrices. This work forms part of my student Ravi Srinivasan's dissertation.