Brownian Motion & Stochastic Calculus
Einstein (1905), Wiener (1923) and Itô (1944-51) turned the random jitter of tiny particles into mathematics for random paths, used from stock pricing to AI image generators.
Open in the interactive tree →In 1827 the botanist Robert Brown saw pollen grains jiggling in water; Einstein showed in 1905 that collisions with water molecules cause it, and Jean Perrin's experiments of 1908-09 confirmed this and helped settle that atoms exist. Norbert Wiener gave the first rigorous mathematical analysis, the Wiener process, in 1923, and Kiyosi Itô published his stochastic integral in 1944 and his change-of-variables formula in 1951, which together allow equations that are driven by random noise. Itô calculus became the tool behind the Black-Scholes option-pricing formula, and in 2020 Song and colleagues described AI image generators as a random process that adds noise to pictures step by step and its reverse that removes it.
Prerequisites
- Statistical Mechanics1877Einstein explained the jitter as collisions with molecules.
- Measure-Theoretic Probability1902-1933Wiener gave the first rigorous mathematical model of the random path.
Unlocks
- Diffusion Models2020Diffusion models are noise-adding and noise-removing stochastic differential equations.
Sources
More in Mathematics · Machine Age
- Hilbert's Problems & Program1900
- Measure-Theoretic Probability1902-1933
- Markov Chains1906
- Zermelo-Fraenkel Set Theory (ZFC)1908-1922
- Abstract Algebra (Noether)1921-1931
- Significance Testing (P-Values)1925-1933
- Gödel's Incompleteness1931
- Computability (Turing)1936