This book is a comprehensive exploration of the interplay between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs). It aims to investigate the influence of geometry on diffusions induced by underlying structures, such as Riemannian or sub-Riemannian geometries, and examine the implications for solving problems in PDEs, mathematical finance, and related fields. The book aims to unify the relationships between PDEs, nonholonomic geometry, and stochastic processes, focusing on a specific condition shared by these areas known as the bracket-generating condition or Hörmander's condition. The main objectives of the book are:To unify the relationship between PDEs, nonholonomic geometry, and stochastic processes by examining the common condition imposed on vector fields in both fields. To explore diffusions induced by underlying geometry, whether Riemannian or sub-Riemannian, and study how curvature affects the diffusion of Brownian movement along curves. To compute heat kernels and fundamental solutions for various operators, using stochastic methods, and analyze their properties. To investigate the dynamics of elliptic and sub-elliptic diffusions on different geometric structures and their applications. To explore the connections between sub-elliptic differential systems and sub-Riemannian geometry. To analyze the dynamics of LC-circuits using variational approaches and establish their relationship with stochastic analysis and geometric analysis. The intended audience for this book includes researchers and practitioners in mathematics, physics, and engineering, who are interested in stochastic techniques applied to geometry and PDEs, as well as their applications in mathematical finance and electrical circuits.