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 ratchet lower temperature than the resistor. Researchers from the University of Twente, the University of Patras in Greece, and the Foundation for Fundamental Research on Matter have constructed a Feynman-Smoluchowski engine which, when not in thermal equilibrium, converts pseudo-Brownian motion into work by means of a granular gas, which is a conglomeration of solid particles vibrated with such vigour that the system assumes a gas-like state. The constructed engine consisted of four vanes which were allowed to rotate freely in a vibrofluidized granular gas. Because the ratchet's gear and pawl mechanism, as described above, permitted the axle to rotate only in