EVENT DETAILS AND ABSTRACT

Title: Theory of Geometric Stable processes
Speaker: Professor Renming Song
Speaker Info: University of Illinois, Urbana-Champaign
Brief Description:
Special Note:
Abstract:

Geometric stable processes have widespread applications in mathematical finance and operations research. In this talk we will discuss some recent progress in the study of symmetric geometric stable processes. Our approach to symmetric geometric stable processes is to realize them as subordinate Brownian motions via geometric stable subordinators. The asymptotic behaviors of the Green function and jumping function near zero of symmetric geometric stable processes exhibit features that are very different from the ones for stable processes. The Green functions behaves near zero as $1/(|x|^d \log2 |x|)$, while the jumping function behaves like $1/|x|^d$. We will also talk about the asymptotic behaviors of the Green function and jumping function of subordinate Brownian motions with iterated geometric stable subordinators. As applications of the asymptotic behaviors, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.