Topics in Stochastic Analysis and Riemannian Foliations

This dissertation contains three research directions. In the first direction, we use rough paths theory to study stochastic differential equations and SPDEs. We first prove convergence and the rate of convergence of the Taylor expansion for the solutions of differential equations driven by $p$-rough paths with $p\u3e2$. The main results are the Castell expansion and the tail estimate for the remainder terms. Our results apply to differential equations driven by continuous centered Gaussian process with finite $2D~\rho-$variation and fBm with $H\u3e1/4$. We then give a new and simple method to get a priori bounds on rough partial differential equations. The technique is based on a weak formulation of the equation and a rough version of Gronwall\u27s lemma. The method is presented on a linear stochastic heat equation. In the second direction, we study stochastic analysis on the horizontal paths space of totally geodesic Riemannian foliations. We first develop Malliavin calculus on the horizontal path space and then prove the quasi-invariance of horizontal Wiener measure. We further prove a Log-Sobolev inequality, the improved Log-Sobolev inequality and the equivalence of two-sided uniform Ricci curvature bounds to functional inequalities. We also obtain concentration and tail estimates. In the third direction, we study Ricci flow on totally geodesic Riemannian foliations. Under the transverse Ricci flow, we prove two types of differential Harnack inequalities for the positive solutions of the heat equation. We also get a time dependent generalized curvature dimension inequality. As consequences, we get parabolic Harnack inequalities and heat kernel upper bounds