Short answers, pulled from the story.
The Scottish botanist Robert Brown first described Brownian motion in 1827 while observing pollen grains from the plant Clarkia pulchella suspended in water under a simple microscope. He confirmed the motion was not biological by repeating the experiment with inorganic particles such as glass and rock dust.
In one of his 1905 papers, Einstein modeled Brownian motion as the result of individual water molecules striking suspended particles. He derived a diffusion equation showing that a particle's displacement is proportional to the square root of elapsed time, not elapsed time itself, and used this relationship to estimate the size of atoms and Avogadro's number.
Jean Baptiste Perrin was awarded the Nobel Prize in Physics in 1926 for what the committee described as his work on the discontinuous structure of matter. His meticulous experiments, published in 1909, confirmed Einstein's predictions and settled the scientific debate over whether atoms and molecules physically exist.
The Wiener process is the formal mathematical description of Brownian motion, named after Norbert Wiener, who provided the first complete and rigorous mathematical analysis of the phenomenon in 1923. It is a continuous-time stochastic process with independent, normally distributed increments and is one of the best-known Levy processes.
The instantaneous velocity of a single Brownian particle was first successfully measured in 2010. Researchers trapped a glass microsphere in air using optical tweezers and verified that the velocity data matched the Maxwell-Boltzmann velocity distribution, confirming the equipartition theorem at microscopic timescales.
In 1900, Louis Bachelier developed the mathematics of Brownian motion in his doctoral thesis titled The Theory of Speculation, prepared under Henri Poincare, to model stock and option price fluctuations. His work was largely unknown until the 1950s, even though it introduced the probabilistic framework later applied to physics.