Brownian Motion

Why Brownian motion matters

• Stochastic calculus. Itô's integral against B(t) is the foundation of every continuous-time stochastic model — solutions to SDEs, martingale representation, the Itô isometry.
• Black–Scholes options pricing. Geometric Brownian motion dS = μS dt + σS dB is the canonical model. Itô's formula on log S gives a closed-form European option price; the 1997 Nobel Prize honored the framework.
• Statistical physics. Einstein's 1905 derivation explained microscopic Brownian motion of pollen via molecular collisions — direct evidence for atoms. Perrin's 1908 experiments measured Avogadro's number from the diffusion law <|B(t)|²> = 2Dt.
• Diffusion and the heat equation. The transition density of B(t) is the heat kernel; Feynman–Kac expresses solutions of parabolic PDEs as Brownian-path averages.
• Random matrix theory. Dyson's Brownian motion describes eigenvalue dynamics; large random matrices have spectra determined by Brownian processes.
• Rough paths and stochastic control. Lyons' rough path theory generalizes integration against Brownian-like signals; applications include filtering, control, and machine learning.
• Modeling. Genetic drift in evolutionary biology, polymer chains in physical chemistry, neural noise in computational neuroscience — Brownian motion is the universal first-order model of fluctuations.

Formal definition

A standard Brownian motion on [0, ∞) is a stochastic process (B(t))_t≥0 on a probability space (Ω, F, P) satisfying:

(1) B(0) = 0 a.s.
(2) For 0 ≤ s < t, B(t) − B(s) ~ N(0, t − s)
(3) For 0 ≤ t₀ < t₁ < ... < tₙ, the increments
    B(t₁) − B(t₀), ..., B(tₙ) − B(tₙ₋₁) are independent
(4) t ↦ B(t, ω) is continuous for almost every ω

Wiener's 1923 construction proved existence — non-trivial because (1)–(4) demand a probability measure on the (uncountable) path space C[0, ∞). Lévy's 1939 construction uses a Schauder basis expansion B(t) = ∑ Z_i ϕ_i(t) where ϕ_i are tent functions and Z_i are i.i.d. standard normals.

Path properties

• Continuous. Sample paths are continuous functions of t, almost surely.
• Nowhere differentiable. For almost every ω, B(t, ω) has no derivative at any t — proved by Paley–Wiener–Zygmund (1933).
• Hölder regularity. Paths are α-Hölder continuous for every α < 1/2, but not for α = 1/2.
• Total variation. Infinite over every subinterval [a, b].
• Quadratic variation. Finite: lim ∑ (B(t_{k+1}) − B(t_k))² = t over partitions of [0, t] with mesh → 0.
• Self-similarity. For any c > 0, (c⁻¹·B(c²t)) is again standard Brownian motion.
• Time reversal. (B(T) − B(T − t))_{0 ≤ t ≤ T} is also Brownian motion on [0, T].
• Local maxima dense. The set of times at which B has a local maximum is countable, dense in [0, ∞).

Quadratic variation: dB·dB = dt

The total variation of a smooth function f on [0, t] is ∫|f'(s)| ds. For Brownian motion this is infinite — the path wiggles too much. But the quadratic variation:

[B, B]_t = lim_{|Π| → 0} ∑_k (B(t_{k+1}) − B(t_k))² = t  (a.s.)

This is finite and deterministic. It's why Itô calculus has a second-order correction: in the Taylor expansion df(B), the term ½ f''(B) (dB)² survives — and (dB)² behaves like dt, not 0. The mantra dB · dB = dt compresses the entire structure.

Itô's formula

For f ∈ C²(ℝ) and B Brownian motion:

f(B(t)) − f(B(0)) = ∫₀ᵗ f'(B(s)) dB(s) + (1/2) ∫₀ᵗ f''(B(s)) ds

The first integral is an Itô stochastic integral; the second is an ordinary Lebesgue integral. The half-second-derivative correction is the entire reason ordinary calculus fails. As an example: f(x) = x². Then f(B(t)) = B(t)², and Itô gives B(t)² = 2 ∫ B dB + t — so the process B(t)² − t is a martingale (it has zero drift). Useful for computing E[τ] and E[B(τ)²] in stopping problems.

Random walk to Brownian motion

Donsker's invariance principle (1951): if Sₙ = X₁ + ... + Xₙ for i.i.d. mean-0, variance-1 random variables, then the rescaled path:

W_n(t) = S_⌊nt⌋ / √n + (nt − ⌊nt⌋)(X_{⌊nt⌋+1}) / √n

converges in distribution to standard Brownian motion B(t) on C[0, 1] as n → ∞. This is the path-space analog of the central limit theorem: sums of small independent shocks become Brownian on the macro scale, regardless of the shocks' distribution (so long as they have finite variance).

Heat equation connection

The transition density p(x, t) = (1/√(2πt)) e^(−x²/(2t)) of B(t) satisfies the heat equation:

∂p/∂t = (1/2) ∂²p/∂x²,    p(x, 0) = δ(x)

More generally, if u(x, t) = E[f(x + B(t))], then ∂u/∂t = ½ Δu. Feynman–Kac extends to time-dependent and space-dependent operators: solutions to ∂u/∂t = ½ Δu + V(x) u with terminal data are expectations of e^(−∫V) f(B(T)) along Brownian paths. PDE problems become path integrals.

Multi-dimensional Brownian motion

An n-dimensional Brownian motion B(t) = (B₁(t), ..., Bₙ(t)) has independent components, each a standard 1-D Brownian motion. Brownian motion is rotationally invariant — its law is unchanged under any orthogonal transformation. In dimension 1 and 2, Brownian motion is recurrent (visits every neighborhood of every point infinitely often); in dimension 3 and higher, it is transient (escapes to infinity, never returns). This is Pólya's theorem in continuous time.

Common misconceptions

• "Just a random walk." Brownian motion is the continuous-time scaling limit; it has no smallest jump and infinite total variation per unit time. A random walk has discrete steps with finite total variation.
• "Differentiable somewhere." Almost surely nowhere differentiable — there is no time t at which B'(t) exists for typical paths.
• "Total variation is finite." Total variation is infinite over any sub-interval. Only the quadratic variation is finite (and equals t).
• "It's a function." It's a probability measure on the space of continuous functions C[0, ∞), induced by Wiener's construction. A "sample path" is one realization drawn from this measure.
• "Average position grows linearly." E[B(t)] = 0 always; it's the standard deviation that grows like √t.
• "Same as Gaussian noise." Brownian motion is the integral of (formal) white noise; white noise itself is a distributional derivative, not a function.
• "It's a Markov chain." Brownian motion has the Markov property in continuous time and is the canonical example of a continuous-state Markov process — but Markov chains have countable state spaces.