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 meander Brownian meander In the mathematical theory of probability, Brownian meander formula_1 is a continuous non-homogeneous Markov process defined as follows: Let formula_2 be a standard one-dimensional Brownian motion, and formula_3, i.e. the last time before "t" = 1 when formula_4 visits formula_5. Then the Brownian meander is defined by the following: In words, let formula_7 be the last time before 1 that a standard Brownian motion visits formula_5. (formula_9 almost surely.) We snip off and discard the trajectory of Brownian motion before formula_7, and scale the remaining part so that it spans a time interval of length 1. The scaling