Short answers, pulled from the story.
Robert Brown observed that pollen grains from Clarkia pulchella moved with a strange jittery energy when placed in water. He found the particles measured about 6 microns and darted erratically without any visible cause.
Albert Einstein linked particle displacement directly to molecular bombardment where invisible water molecules struck larger pollen grains from all sides. These collisions happened roughly 10^14 times per second for each particle causing constant changes in force due to uneven hits.
Jean Perrin conducted experiments starting in 1908 to confirm Einstein's theoretical equations regarding granules of gamboge moving against gravity. His data matched predictions perfectly and earned him the Nobel Prize in Physics in 1926.
Louis Bachelier introduced the mathematics of stochastic analysis in his 1900 doctoral thesis titled The Theory of Speculation. Norbert Wiener provided the first complete and rigorous mathematical analysis in 1923 for what became known as the Wiener process.
Smoluchowski published a one-dimensional model in 1906 that assumed collisions always imparted the same magnitude of velocity change. Arnold Sommerfeld noted that the numerical coefficient of Einstein differed from Smoluchowski's by exactly 27/64.