Brownian Paths (1D) with √t Envelope
2D Brownian Motion
Quadratic Variation [W]_t ≈ t
Distribution at Fixed Time
Running Maximum & Reflection Principle
The Wiener Process
A standard Wiener process W(t) satisfies: W(0)=0, increments W(t)−W(s) ~ N(0,t−s) for t>s, and increments over non-overlapping intervals are independent.
Key properties: E[W(t)] = μt (with drift), Var[W(t)] = σ²t, so standard deviation grows as σ√t. Paths are continuous but nowhere differentiable almost surely (Lévy 1940).
Quadratic variation: [W]_T = lim_{|Π|→0} Σ(W(tᵢ)−W(tᵢ₋₁))² = T almost surely. This is why dW² = dt in Itô calculus — a fundamental departure from classical calculus.
Connection to heat equation: the transition density of Brownian motion satisfies ∂p/∂t = ½∂²p/∂x². Brownian motion IS the fundamental solution (heat kernel) of the heat equation.
Reflection principle: P(max_{0≤s≤t} W(s) ≥ a) = 2P(W(t) ≥ a) for a > 0. The maximum of Brownian motion has the same distribution as |W(t)|.