Brownian Motion - Riemannian Manifold

Riemannian Manifold

The infinitesimal generator (and hence characteristic operator) of a Brownian motion on Rn is easily calculated to be ½Δ, where Δ denotes the Laplace operator. This observation is useful in defining Brownian motion on an m-dimensional Riemannian manifold (M, g): a Brownian motion on M is defined to be a diffusion on M whose characteristic operator in local coordinates x_i, 1 ≤ i ≤ m, is given by ½Δ_LB, where Δ_LB is the Laplace–Beltrami operator given in local coordinates by

where = −1 in the sense of the inverse of a square matrix.