Brownian motion is the most basic diffusion process on Euclidean space. The goal of this talk is to understand diffusion on spaces beyond Euclidean space. While Brownian motion on manifolds has been extensively studied, expanding this construction to stratified spaces proves challenging. Sturm introduced an approach for constructing diffusion processes on metric measure spaces, relying on the theory of Dirichlet forms and their well-established connection to Markov processes. These forms generate diffusion only if the space satisfies a geometric condition known as the "measure contraction property," which can be viewed as a regularity condition for measures. We explore the geometric condition in the context of Whitney stratified spaces, equipped with the Riemannian distance and a volume measure. I will talk about the relationships between diffusion and Dirichlet forms and their generalization to stratified spaces. No prior knowledge of Dirichlet forms and Markov processes is assumed. This is ongoing work with Celia Hacker, Markus Pflaum and Sayan Mukherjee.
Brownian Motion on Singular Spaces Sponsored by the Meyer Fund
Nov. 15, 2023 4:40pm (MATH 3…
Grad Student Seminar
Jon Kim (University of Colorado)
X
When working with curves and surfaces, it's much nicer to work with them when they are smooth. Alas, we can't always have these nice assumptions, so we gotta ask when smoothness fails and whether if we can resolve it. Well, we know how to identify singularities, but if we can resolve them, we then gotta figure out how to do it in a manner where we can preserve information away from it. Thankfully, it turns out that we do have such a method called blowing-up. This talk will be an introduction to blow-ups in an illustrated manner to via worked examples with and (cartoon)-pictures.
Brownian Motion on Singular Spaces