Brownian Motion

Mathematical Modeling · Axiom Academy

LESSON Brownian Motion - Mathematical Modeling Unit 4: Probabilistic Models - Mathematical Modeling Brownian motion (also called the Wiener process) is a continuous-time stochastic process that serves as the fundamental building block for modeling random phenomena in continuous time. It is the continuous-time limit of random walks. "The mathematical idealization of random, continuous movement." Brownian motion is central to: Financial mathematics (stock price modeling, option pricing) Physics (diffusion, thermal fluctuations) Biology (molecular motion, population genetics) Stochastic calculus and differential equations Signal processing and filtering theory The development of Brownian motion spans physics, biology, and mathematics, involving several brilliant minds. Scottish botanist Robert Brown observed the erratic motion of pollen grains suspended in water under a microscope. Initially thought to be a sign of life, he later observed the same motion in inorganic particles, establishing its physical (not biological) nature. Einstein provided the first theoretical explanation, showing that Brownian motion results from the bombardment of particles by countless water molecules. His paper predicted that mean squared displacement grows linearly with time: . French physicist Perrin experimentally verified Einstein's predictions, providing definitive evidence for the existence of atoms and molecules. He won the 1926 Nobel Prize partly for this work.

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