Fractional Brownian motion stationary and self-similar increments. It has stationary increments: For "H" > ½ the process exhibits long-range dependence, Sample-paths are almost nowhere differentiable. However, almost-all trajectories are locally Hölder continuous of any order strictly less than "H": for each such trajectory, for every "T" > 0 and for every "ε" > 0 there exists a (random) constant "c" such that for 0 < "s","t" < "T". With probability 1, the graph of "B"("t") has both Hausdorff dimension and box dimension of 2−"H". As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional
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