Rotational Brownian motion Rotational Brownian motion Rotational Brownian motion is the random change in the orientation of a polar molecule due to collisions with other molecules. It is an important element of theories of dielectric materials. The polarization of a dielectric material is a competition between torques due to the imposed electric field, which tend to align the molecules, and collisions, which tend to destroy the alignment. The theory of rotational Brownian motion allows one to calculate the net result of these two competing effects, and to predict how the permittivity of a dielectric material depends on the strength and frequency of the
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