Stochastic Analysis On Manifold And Its Application

As a model of many geometry and physics problems,stochastic analysis on manifold has been extensively studied.This thesis discusses the diffusion process on manifold and its application in geometry and machine learning.More precisely,when the manifold has boundary,by combining the reflected Brownian motion and a smooth approximation technique,a new formula for the multiplicative functional of the heat equation for differential forms on manifolds with boundary can be given.Through this formula,we can give a simple proof of the Gauss-bonnet theorem by asymptotically expanding the multiplicative functional under the Brownian bridge measure.The first probabilistic proof of Gauss-bonnet theorem with boundary relies heavily on Excursion theory and Malliavin Calculus.However,our method is solely based on Ito's formula,smooth approximation and stochastic differential equation of the Brownian bridge.As to the applications,we apply the diffusion on manifolds to building the exchangeable pairs,which is an essential ingredient in the infinitesimal Stein 's method.Our method works for general Gibbs measure on manifolds,which extends results of the Haar measure on the orthogonal group in previous literature.On the other hand,we propose the power-law diffusion to modeling the stochastic gradient descent algorithm(SGD).By applying the synchronized coupling technique,we are able to analyze important properties of the discretization of the power-law diffusion and its continuous limit,including the marginal distribution and the first exit time.