Stochastic analysis on Riemannian manifolds without boundary has been well established. However, the analysis for reflecting diffusion processes and sub-elliptic diffusion processes is far from complete. This book contains recent advances in this direction along with new ideas and efficient arguments, which are crucial for further developments. Many results contained here (for example, the formula of the curvature using derivatives of the semigroup) are new among existing monographs even in the case without boundary.
Diffusion Processes on Riemannian Manifolds; Reflecting Diffusion Processes on Riemannian Manifolds with Boundary; Coupling and Applications; Harnack Inequalities and Applications; Functional Inequalities and Applications; Formulae for the Curvature and Second Fundamental Form; Equivalent Semigroup Inequalities for the Lower Bounds of Curvature and Second Fundamental Form; Modified Curvature and Applications; Robin Semigroup and Applications; Stochastic Analysis on the Path Space Over Manifolds with Boundary; Subelliptic Diffusion Processes.