Brownian Motion

Stochastic paths, the Wiener process, and diffusion

Paths (N) 10

Diffusion (σ) 0.50

Drift (μ) 0.00

Time Steps 200

W(t): E[W(t)]=μt, Var[W(t)]=σ²t, W(t)~N(μt, σ²t)

Each path is a realization of a Wiener process with drift μ and diffusion σ. Increments dW = μ·dt + σ·dB are independent Gaussian random variables. The right panel shows the distribution of path endpoints — it converges to N(μT, σ²T) as N→∞. The variance grows linearly with time (diffusion), which distinguishes Brownian motion from ballistic motion.