Shuwen Lou : Brownian Motion with Varying Dimension

Posted by Cheng Ouyang , part of the Statistics and Data Science Seminar.

At

Jan. 22, 2014, 4 p.m.

In

SEO 636

Abstract

Think of installing an infinite pole on top of the ground. We want to model the random movement of an ant on this space. However, as we know, the standard 2-dimensional Brownian motion does not hit a single, which means that once the ant is on the ground, it will never have the chance to climb up the pole. We fix this problem by defining Brownian motion with varying dimension on this state space as a darning process whose rigorous definition will be introduced in the talk. The main results are about global two-sided heat kernel estimates for such processes. We will see from the heat kernel estimates that these processes embody both 1-dimensional property and 2-dimensional property, which depends not only on the regions of the points but also on time.