Metric graphs are graphs where each edge is assigned a length and associated with a closed interval of that length. The study of Brownian motions on metric graphs has garnered significant interest in recent years due to wide-ranging applications in fields like mathematical physics and biology. This talk will provide a gentle introduction to the study of Brownian motions on metric graphs from the ground up, work through some simple examples of metric trees, and compute transition densities of Brownian motions on them. Finally, I will give an overview of some important results in the study of Brownian motions on infinite symmetric metric trees and outline some minor results pertaining to Brownian motions on metric trees that were obtained during my time at the Indiana University Mathematics REU in Summer 2023.
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