Brownian motion on homogeneous space and control systems via Riemannian submersion using mean curvature driftdissertationsHuang, Ching-PengcreatorHassett, BrendanReaderMenon, GovindAdvisorDarbon, JérômeReaderBrown University. Department of Applied Mathematicssponsor2022vi, 48 p.born digitalThesis (Ph. D.)--Brown University, 2022thesesWe begin by asking how to write down Brownian motion on certain quotient Riemannian manifolds. The projection formula for the Laplace-Beltrami operator through Riemannian submersions indicates the Brownian motion on the total space gains a drift term from the mean curvature flow of fibers through the submersion. With this key fact, in the first chapter, we write down formula for Brownian motion on several Riemannian homogeneous spaces, in particular symmetric spaces, using their common embedding coordinates in matrix form. Moreover, we study the Bures-Wasserstein geometry on positive semidefinite matrices and write down a formula of its Brownian motion via computing the aforementioned mean curvature flow. In the second chapter, we investigate the idea of utilizing the mean curvature flow on the Bures-Wasserstein geometry by altering the metric, deriving a Lie theoretical invariant control systemStochastic processesGeometry, RiemannianLie groupscontrol systemrandom matrixEnglishRPB20220706