Brownian bridge of the Brownian bridge based on the Brownian motion is, for "t" ∈ [0, T] Conversely, for "t" ∈ [0, ∞] The Brownian bridge may also be represented as a Fourier series with stochastic coefficients, as where formula_9 are independent identically distributed standard normal random variables (see the Karhunen–Loève theorem). A Brownian bridge is the result of Donsker's theorem in the area of empirical processes. It is also used in the Kolmogorov–Smirnov test in the area of statistical inference. A standard Wiener process satisfies "W"(0) = 0 and is therefore "tied down" to the origin, but other points are not
Reflected Brownian motion steady state RBMs. Feller described possible boundary condition for the process Reflected Brownian motion In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt. A "d"–dimensional reflected Brownian motion "Z" is a stochastic process on formula_1 uniquely defined by where "X"("t") is an unconstrained Brownian motion and with "Y"("t") a "d"–dimensional vector where The reflection matrix describes boundary behaviour. In the interior of