### Shuwen Lou (University of Washington)

Title: Brownian Motion with Varying Dimension

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. The main results are about global two-sided heat kernel estimates. 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.