Seminar Probabilit├ęs -- Moments, rough initial data and intermittency for SPDEs

11/26/2015 - 16:30
11/26/2015 - 17:30
Le Chen (Kansas University)
Burnside Hall, Rm. 920 (McGill University)

Intermittency refers to the property that solution exhibits both tall peaks and low valleys. Zeldovich et al. observed that Intermittency is a very universal phenomenon which occurs practically irrespective of detailed properties of the background instability in a random medium provided only that the random field is of multiplicative type … Our goal is to turn this qualitative statement into quantitative theorems for various stochastic partial differential equations (SPDEs) subject to some noise of multiplicative type. In particular, I will first present, in this talk, some results on the stochastic heat equation (SHE). A key tool in this study is a sharp moment formula, which enables us to establish existence of a random field solution starting from measure-valued initial data. Moreover, when initial data is localized, using this moment formula, we are able to establish some exact propagation speeds of these tall peaks, the so-called intermittency fronts or growth indices, certain quantities defined and first studied by Conus and Khoshnevisan. Then I will show that techniques used to study SHE can be applied to various other SPDEs, such as the stochastic wave equation, SHE with the Laplacian replaced by a fractional Laplacian, and SHE on R^d with a spatially colored noise. This talk is based on some joint works with Robert C. Dalang and Kunwoo.

Last edited by on Mon, 11/23/2015 - 17:32