Brownian meander factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point formula_5. The transition density formula_12 of Brownian meander is described as follows: For formula_13 and formula_14, and writing we have and In particular, i.e. formula_19 has the Rayleigh distribution with parameter 1, the same distribution as formula_20, where formula_21 is an exponential random variable with parameter 1. Brownian meander In the
Fractional Brownian motion Hypergeometric functions are part of the GNU scientific library. Fractional Brownian motion In probability theory, fractional Brownian motion (fBm), also called a fractal Brownian motion, is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. fBm is a continuous-time Gaussian process "B"("t") on [0, "T"], which starts at zero, has expectation zero for all "t" in [0, "T"], and has the following covariance function: where "H" is a real number in (0, 1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. The Hurst exponent describes the raggedness