Brownian Motion

Brownian Motion is a stochastic process W(t) which has the following four properties:

1] W(0) = 0, 

2] almost surely, the trajectory of W(t) is continuous;

3] W(t) has independent increments: for any moments of time s < t < u, random variables W(t) - W(s) and W(u) - W(s) are independent;

4] W(t) has stationary increments: for any moments of time s < t and positive shift h, random variables W(s+h) - W(s) and W(t+h) - W(t) have the same distribution. 

It follows from properties 3, 4 and the Central Limit Theorem that any finite-dimensional distributions of a Brownian motion are Gaussian (normal). For example, in the definition above, random variables W(t) - W(s), W(u) - W(s), W(s+h) - W(s) and W(t+h) - W(t) are jointly Gaussian.

Brownian motion is also known as "standard Wiener process".