|Dept.||University of Utah|
Intermittency occurs in many areas of science such as turbulence, cosmology, neuroscience and finance. Intuitively speaking, intermittency means that tall peaks in many different length scales can develop, usually as time grows to infinity.
In the first part of this talk, we consider a large family of intermittent stochastic partial differential equations (SPDEs), which deal with generators of Levy processes on LCA groups. Instead of looking at a large time behavior, we investigate nonlinear noise excitation on those SPDEs. We show a surprising result that there is a near-dichotomy: “Semi-discrete” equations are nearly always far less excitable than “continuous” equations.
In the second part of this talk, we consider various parabolic Anderson models (PAM), which exhibit intermittency and provide Hopf-Cole solutions to KPZ equations. We show that tall peaks of the solutions to PAM are multi-fractal in macroscopic scale. Some of the examples include stochastic fractional heat equations.
This is based on joint works with Davar Khoshnevisan and Yimin Xiao.