Brownian motion stochastic process models. There exist both simpler and more complicated stochastic processes which in extreme ("taken to the limit") may describe the Brownian Motion (see random walk and Donsker's theorem). The Roman Lucretius's scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of Brownian motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms: "Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of
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