Rotational Brownian motion (astronomy) that the rotation axis of the binary is oriented at angle θ at time "t". The evolution equation for "F" is formula_17 If <Δξ>, "a", ρ and σ are constant in time, this becomes formula_18 where μ = cos θ and τ is the time in units of the relaxation time "t", where formula_19 The solution to this equation states that the expectation value of μ decays with time as formula_20 Hence, "t" is the time constant for the binary's orientation to be randomized by torques from field stars. Rotational Brownian motion was first discussed in the context of binary
Brownian motion continuous stochastic process on a probability space (Ω, Σ, P) taking values in R. Then the following are equivalent: The infinitesimal generator (and hence characteristic operator) of a Brownian motion on R is easily calculated to be ½Δ, where Δ denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. 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 formula_46 in local coordinates