Rotational Brownian motion (astronomy) supermassive black holes at the centers of galaxies. Perturbations from passing stars can alter the orbital plane of such a binary, which in turn alters the direction of the spin axis of the single black hole that forms when the two coalesce. Rotational Brownian motion is often observed in N-body simulations of galaxies containing binary black holes. The massive binary sinks to the center of the galaxy via dynamical friction where it interacts with passing stars. The same gravitational perturbations that induce a random walk in the orientation of the binary, also cause the binary to shrink, via the gravitational
Fractional Brownian motion stochastic integrals". In general though, unlike integrals with respect to regular Brownian motion, fractional stochastic integrals are not semimartingales. Just as Brownian motion can be viewed as white noise filtered by formula_8 (i.e. integrated), fractional Brownian motion is white noise filtered by formula_9 (corresponding to fractional integration). Practical computer realisations of an "fBm" can be generated, although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an "fBm" process. Three realizations are shown below, each with 1000 points of an "fBm" with Hurst parameter 0.75. Realizations of three different