ABSTRACT
This chapter is concerned with the design of Brownian motion and diffusion processes in local coordinate systems. It discusses the Doeblin-Ito formula associated with these diffusion processes in Riemannian manifolds. The chapter illustrates Brownian motion on the sphere, on the torus, and on the simplex. It also discusses the Brownian motion on the orbifold. The positive orthant of the sphere is in bijection with the p-simplex. The unit circle can be described in terms of the polar coordinates mapping. The 2-sphere can be parametrized by the spherical coordinates mapping. The chapter provides an illustration of a realization of a Brownian motion on the torus.
