Brownian Motion: From Physics to Stochastic Math

Definition and physical setting of Brownian motion

Brownian motion is the random motion of particles suspended in a medium such as a liquid or a gas. In its traditional mathematical description, it corresponds to the Wiener process. Physically, it appears as irregular fluctuations of a particle’s position within a small region of the fluid, followed by relocation to another region; this repeated pattern reflects a fluid in thermal equilibrium at a given temperature, with no preferred flow direction and with overall linear and angular momenta remaining null over time. The phenomenon is rooted in microscopic molecular impacts: the particle is bombarded by molecules whose directions and timings change constantly, producing an apparently random trajectory. The motion is consistent with thermal equilibrium and the equipartition theorem, since the kinetic energies associated with molecular motion (including rotations and vibrations) contribute to the fluid’s internal energy. Brownian motion is named after Robert Brown, who first observed the jittery motion of pollen grains in water under a microscope in 1827.

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Wiener process as mathematical Brownian motion

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