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What is Brownian Motion?
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The basic definition is $$ $$ $W_t$, $t \geq 0$ is a Brownian motion if $W_0 = 0$ and $W_t - W_s$ is a normally distributed random variable with mean zero and variance equal to $t-s$. $W_t - W_s$ is independent of $W_r$ for all $r \leq s$.