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In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

The standard Wiener process $W_t$ is characterized by four facts:

  1. $W_0$ = 0
  2. $W_t$ is almost surely continuous.
  3. $W_t$ has independent increments.The condition that it has independent increments means that if $0\leq s_1\leq t_1\leq s_2\leq t_2$ then $W_{t_1}-W_{s_1}$ and $W_{t_2}-W_{s_2}$ are independent random variables.
  4. $W_{t}-W_{s}\tilde{} N(0,t-s)$