ABSTRACT
This chapter is concerned with the construction of Brownian motion and more general diffusion processes evolving in constraint type manifolds. These stochastic processes are defined in terms of projection of Euclidian Brownian motions and multidimensional diffusions adjusted by mean curvature drift functions. The chapter discusses the Doeblin-Ito formula associated with these manifold valued diffusions in the ambient space. It provides a brief discussion to the Stratonovitch differential calculus. The chapter illustrates these probabilistic models with a variety of concrete examples, including Brownian motion and diffusions on the sphere, the cylinder, the simplex or the more general orbifold. It describes some terminology that is used frequently in geometry and differential calculus.
