Brownian motion are independent formula_45 random variables. This representation can be obtained using the Karhunen–Loève theorem. The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant. The time evolution of the position of the Brownian particle
Brownian bridge Brownian bridge A Brownian bridge is a continuous-time stochastic process "B"("t") whose probability distribution is the conditional probability distribution of a Wiener process "W"("t") (a mathematical model of Brownian motion) subject to the condition (when standardized) that "W"(T) = 0, so that the process is pinned at the origin at both "t=0" and "t=T". More precisely: The expected value of the bridge is zero, with variance formula_2, implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance of "B"("s") and "B"("t") is "s"(T − "t")/T if "s" < "t". The