Brownian motion
Brownian motion \(W(t)\) is a continuous-time stochastic process used to represent irregular random fluctuations.
It satisfies \(W(0)=0\), has independent increments, and for \(0\le s<t\),
\[W(t)-W(s)\sim N(0,t-s).\]Its sample paths are continuous but almost surely nowhere differentiable.
Key idea. Brownian motion changes continuously in time, although its path is random and extremely irregular.