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Brownian motion exhibits remarkable structural properties beyond its definition. It is self-similar (scaling invariance: cW(t/c²) is again a Brownian motion), has unbounded variation but finite quadratic variation equal to t, satisfies the strong Markov property, and obeys a reflection principle. The quadratic variation [W,W]_t = t is the single most consequential property for stochastic calculus — it is the reason Itô's formula has an extra term compared to the classical chain rule.
Beyond its four defining properties, Brownian motion possesses a constellation of structural features that make it uniquely tractable and deeply connected to analysis. The most important of these is quadratic variation. For a partition 0 = t₀ < t₁ < ... < tₙ = T of [0,T], the quadratic variation is the limit of Σ(W(tᵢ) - W(tᵢ₋₁))² as the mesh goes to zero. Each squared increment (W(tᵢ) - W(tᵢ₋₁))² has mean tᵢ - tᵢ₋₁ and variance 2(tᵢ - tᵢ₋₁)², so the sum has mean T and variance that goes to zero — it converges in L² to T. This deterministic quadratic variation [W,W]_T = T, summarized as the heuristic (dW)² = dt, is the engine of Itô calculus.
Self-similarity (scaling invariance) states that (1/√c)W(ct) is again a standard Brownian motion for any c > 0. Brownian motion looks statistically identical at every timescale — zoom into a small segment and rescale, and you see the same statistical object. This fractal character is reflected in the Hausdorff dimension of 3/2 for the graph of t ↦ W(t). Related symmetries include time inversion (tW(1/t) is a Brownian motion) and the reflection principle (|W(t)| or W reflected at its maximum relate the distribution of the running maximum to the process itself). The reflection principle yields the distribution of the maximum: P(max_{s≤t} W(s) ≥ a) = 2P(W(t) ≥ a) for a > 0.
The strong Markov property extends the ordinary Markov property from deterministic times to stopping times: given the process at a stopping time τ, the future process W(τ + t) - W(τ) is an independent Brownian motion. This is essential for analyzing first-passage times and boundary problems. Combined with the reflection principle, it implies that the first hitting time T_a = inf{t : W(t) = a} has an inverse Gaussian distribution with E[T_a] = ∞ — Brownian motion will eventually hit any level, but the expected time to do so is infinite.
The contrast between total variation (infinite) and quadratic variation (finite) determines the entire character of stochastic calculus. Smooth functions have finite total variation and zero quadratic variation; Brownian motion has infinite total variation but deterministic quadratic variation equal to t. This places Brownian paths in a precise regularity class: too rough for ordinary calculus (which assumes zero quadratic variation), but regular enough for the Itô integral (which requires finite quadratic variation). Every major result in stochastic calculus — Itô's formula, the Girsanov theorem, the martingale representation theorem — traces back to this fundamental property.