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 surface Brownian surface A Brownian surface is a fractal surface generated via a fractal elevation function. For instance, in the three-dimensional case, where two variables "X" and "Y" are given as coordinates, the elevation function between any two points ("x", "y") and ("x", "y") can be set to have a mean or expected value that increases as the vector distance between ("x", "y") and ("x", "y"). There are, however, many ways of defining the elevation function. For instance, the fractional Brownian motion variable may be used, or various rotation functions may be used to achieve more natural looking surfaces. As with