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
Brownian motion moment of probability of displacement formula_2, is interpreted as mass diffusivity "D": Then the density of Brownian particles "ρ" at point "x" at time "t" satisfies the diffusion equation: Assuming that "N" particles start from the origin at the initial time "t"=0, the diffusion equation has the solution This expression (which is a normal distribution with the mean formula_12 and variance formula_13 usually called Brownian motion formula_14) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to