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 motor at the nanometer scale. The term "Brownian motor" was originally coined by Peter Hänggi in 1995: A distinct feature of a Brownian motor is—in contrast to a molecular motor—that the output response is typically coupled only loosely to the input perturbation and action of fluctuations. In biology, many protein-based molecular motors in the cell may in fact be Brownian motors. These molecular motors convert the chemical energy present in ATP into mechanical energy. One example of a Brownian motor would be an ATPase motor that hydrolyzes ATP to generate fluctuating anisotropic energetic potentials. The anisotropic potentials along the path would