Functional Inequalities for Hypoelliptic Diffusions Using Probabilistic and Geometric Methods

We study gradient bounds and other functional inequalities related to hypoelliptic diffusions. One of the key techniques in our work is the use of coupling of diffusion processes to prove gradient bounds. We also use generalized $\Gamma$-calculus to prove various functional inequalities. In this dissertation we present two research directions; gradient bounds for harmonic functions on the Heisenberg group, and gradient bounds for the heat semigroup generated by Kolmogorov type diffusions. For the first research direction, we construct a non-Markovian coupling for Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, estimates for the CDF of the coupling time and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally, we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic sub-Laplacian which is the generator of Brownian motion in the Heisenberg group. In particular, we prove the well known Cheng-Yau inequality and a Caccioppoli-type inequality on the Heisenberg group. For the second research direction, we study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov-type operators. Unlike the first research direction, the focus is on two different methods: coupling techniques and generalized $\Gamma$-calculus techniques. We discuss the advantages and drawbacks of each of these methods. For the coupling technique, we use a coupling by parallel transport (or synchronous coupling) to induce a coupling on the Kolmogorov type diffusions. In the $\Gamma$-calculus approach, we will prove a new generalized curvature dimension inequality to study various functional inequalities