Regular Brownian Motion

Regular Brownian motion, or simply Brownian motion, is named in honour of the Scottish botanist Robert Brown (1773–1858) who, while using his microscope to observe pollen grains floating in water, noticed that they underwent rapid irregular motions. Brown found that other small particles also exhibited these seemingly unpredictable movements when placed on the water surface and he reasoned that the movement must be due to physical causes. We now know that the highly irregular motion of suspended particles at the water surface is due to their bombardment by the water molecules. Brownian motion is therefore a macroscopic manifestation of the molecular motion of the liquid. If we release a group of particles in a fluid at a specific location the action of the bombarding molecules in the liquid will cause the particles to spread out, or diffuse, through time. Molecular diffusion simulations based on Brownian motion are extensively used in science and engineering to model diffusion processes in both solid and fluid media.
Definition (Brownian motion, B(t)). Brownian motion is a function B(t), defined for equally-spaced time steps $\delta t$, such that all increments $\delta B(t)$ are independent, isotropic, and random. Independent means that the value of the current increment does not affect the value of the next. Isotropic means that the increments are equally likely to occur in all directions. Random means that the values of the increments are unpredictable.
Instead of thinking about Brownian motion in terms of increments $\varepsilon$ in the position of the particle,Wiener considers the random function B(t), which describes the position itself.