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
Reflected Brownian motion all "t" ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for "μ" < 0) when taking t → ∞ an exponential distribution For fixed "t", the distribution of "Z(t)" coincides with the distribution of the running maximum "M(t)" of the Brownian motion, But be aware that the distributions of the processes as a whole are very different. In particular, "M(t)" is increasing in "t", which is not the case for "Z(t)". The heat kernel for reflected Brownian motion at formula_9: formula_10 For the plane above formula_11 The stationary distribution of a reflected Brownian motion