Stochastic areas, Horizontal Brownian Motions, and Hypoelliptic Heat Kernels

TL;DR

This monograph explores stochastic area functionals of Brownian motions and their heat kernels on Lie groups and manifolds, highlighting the interplay between stochastic calculus, geometry, and symmetric spaces.

Contribution

It provides a comprehensive, self-contained treatment of stochastic areas and heat kernels, with concrete examples illustrating deep geometric and probabilistic interactions.

Findings

Analysis of stochastic area functionals on Lie groups

Explicit heat kernel formulas in sub-Riemannian settings

Connections between stochastic calculus and geometric structures

Abstract

The monograph is devoted to the study of stochastic area functionals of Brownian motions and of the associated heat kernels on Lie groups and Riemannian manifolds. It is essentially self-contained and as such can serve as a textbook on the theory of Brownian motions and horizontal Brownian motions on manifolds. Emphasis is put on concrete examples which allows us to concretely illustrate the rich and deep interactions between stochastic calculus, Riemannian and sub-Riemannian geometry, the theory of complex and quaternionic symmetric spaces and random matrices.