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
Brownian motion motion can be measured as "v" = Δ"x"/Δ"t", when Δ"t" « "τ", where "τ" is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle. In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity "V" of the massive object, of mass "M", is related to the rms velocity formula_34 of the background