Brownian Motion:

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Brownian motion B(t) is a stochastic process with the following properties:

• B(t)-B(s) for t>s is independent of the past B(u) for s>=u.

• B(t)-B(s) has Normal distribution with mean 0 and variance t-s. (Hence, B(t)~N(0,t))

• B(t) is continuous

Examples of phenomena that appear to behave in this manner include stock market time-series data,​ particle motion, star motion within galaxies, etc.

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Below we show some numerical simulations of Brownian motion in 1-dimension:

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